special topics in numerics -...

SPECIAL TOPICS IN NUMERICS(Nonlinear Problems, Mixed Methods and Adaptivity)

Rolf Rannacher

Institute of Applied Mathematics

Heidelberg University

Lecture Notes SS 2016

July 29, 2016

ii

Address of Author:

Institute of Applied MathematicsHeidelberg UniversityIm Neuenheimer Feld 205 (MATHEMATIKON)D-69120 Heidelberg, Germany

[email protected]

http://www.uni-heidelberg.de/numerik

Contents

0 Introduction 1

0.1 List of topics covered in this course (and its continuation) . . . . . . . . . . . . . 1

0.2 Basic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

0.3 Basics of finite element method for linear problems . . . . . . . . . . . . . . . . . 2

0.4 Useful technics of mathematical analysis in the finite element method . . . . . . 3

0.4.1 Duality arguments (“Aubin-Nitsche trick”) . . . . . . . . . . . . . . . . . 4

0.4.2 Inverse estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

0.4.3 Local integral inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

0.4.4 Lax-Milgram Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

0.5 Exercises (for refreshing the knowledge of some preparatory material) . . . . . . 5

1 Some Special Types of Nonlinear Problems 9

1.1 Examples of nonlinear problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.1 Minimization problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.2 Nonlinear diffusion-reaction-transport problems . . . . . . . . . . . . . . . 10

1.1.3 Von Karman model in plate bending theory . . . . . . . . . . . . . . . . . 12

1.1.4 Eigenvalue problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Convex minimization problems and variational inequalities . . . . . . . . . . . . . 13

1.2.1 Approximation of abstract variational inequalities . . . . . . . . . . . . . 16

1.2.2 Application to obstacle and Signorini problem . . . . . . . . . . . . . . . . 18

1.3 The minimal surface problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.3.1 Finite element approximation . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.4 Problems of monotone type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.4.1 An abstract error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.4.2 Application to the p-Laplace problem . . . . . . . . . . . . . . . . . . . . 29

1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2 General Quasilinear Elliptic Problems 39

2.1 Quasi-linear problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.1.1 Finite element discretization . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.1.2 Auxiliary L∞-stability estimates for the linearized problems . . . . . . . . 46

2.2 Solution of the discretized problems . . . . . . . . . . . . . . . . . . . . . . . . . 54

iii

iv CONTENTS

2.2.1 A brief survey of iterative solution methods . . . . . . . . . . . . . . . . . 54

2.2.2 The Newton method in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.2.3 The Newton method in function space . . . . . . . . . . . . . . . . . . . . 61

2.2.4 The projective Newton method . . . . . . . . . . . . . . . . . . . . . . . . 65

2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3 The (stationary) Navier-Stokes System 71

3.1 The stationary Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . . 71

3.1.1 The Stokes operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.1.2 Existence result for the Navier-Stokes problem . . . . . . . . . . . . . . . 74

3.1.3 Iterative solution schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.2 Finite element discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.2.1 General “Stokes elements” . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.2.2 Stabilized Stokes elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.2.3 Navier-Stokes problem: the small-data case . . . . . . . . . . . . . . . . . 104

3.2.4 Transport stabilization for more general data . . . . . . . . . . . . . . . . 108

3.2.5 A prototypical example in 1D . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.2.6 Treatment of nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

3.2.7 Solution of linear discrete problems . . . . . . . . . . . . . . . . . . . . . . 116

3.2.8 Schur complement methods . . . . . . . . . . . . . . . . . . . . . . . . . . 117

3.2.9 Multigrid method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

3.3 A nested solution scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4 Adaptivity (to come) 133

5 Mixed Methods (to come) 135

Bibliography 137

0 Introduction

0.1 List of topics covered in this course (and its continuation)

– Basics of FEM (Literature: Lecture Notes Rannacher [3, 51], Books of Braess [25], Ciarlet[26], Quarteroni&Valli [28], Strang&Fix [29]), Johnson [32], Brenner&Scott [34]).

– Special types of non-linear problems: variational inequalities, obstacle problem, minimalsurface problem, problems of monotone type (Literature: Book of Ciarlet [26], Article ofRannacher [45]).

– General quasi-linear elliptic boundary value problems, Newton method, mesh independentconvergence (Literature: Articles of Frehse&Rannacher [39, 40], Dobrowolski &Rannacher[38], Rannacher [45, 47, 48, 52], Rannacher&Scott [53]).

– Mixed FE schemes for second- and fourth-order problems (in the 2nd part; Literature:Books of Brezzi &Fortin [33], Ciarlet [26], Brenner&Scott [34], Article of Rannacher [46]).

– The (stationary) Navier-Stokes equations, linearization procedures, stable Stokes elements,pressure and transport stabilization, solution of linear sub-systems, nested solution ap-proaches (Literature: Lecture Notes of Rannacher [4], Books of Temam [19], Galdi [7],Girault&Raviart [27], Brenner&Scott [34], Articles of Heywood&Rannacher [42], Hey-wood, Rannacher&Turek [43], Rannacher&Turek [54], Rannacher [49, 50, 51]).

– Adaptivity: a posteriori error estimation, mesh-refinement strategies, simultaneous con-trol of mesh refinement and algebraic iteration (in the 2nd part; Literature: LectureNotes of Rannacher [51, 3], Book of Bangerth&Rannacher [24], Articles of Becker, John-son&Rannacher [36], Becker &Rannacher [37], Heuveline&Rannacher [41], Meidner, Ran-nacher&Vihharev [44], Rannacher&Vihharev [55]).

0.2 Basic notation

We have to deal with scalar or vector-valued functions u = u(x) for arguments x ∈ Rn . For

derivatives of differentiable functions, we use the notation

∂xu :=∂u

∂x, ∂2xu :=

∂2u

∂2x, . . . , ∂iu :=

∂u

∂xi, ∂2iju :=

∂2u

∂xi∂xj, . . . ,

and analogously also for higher-order derivatives. With the nabla operator ∇ the “gradient”of a scalar function and the “divergence” of a vector function are written as gradu = ∇u :=(∂1u, ..., ∂du) and divu = ∇ · u := ∂1u1 + ... + ∂dud , respectively. For a vector β ∈ R

d thederivative in direction β is written as ∂βu := β · ∇u . Combination of gradient and divergenceyields the “Laplacian operator”

∇ · ∇u = ∆u = ∂21u+ ...+ ∂2du.

The symbol ∇mu denotes the “tensor” of all partial derivatives of order m of u , i. e., in twodimensions u = u(x1, x2) , ∇2u = (∂i1∂

j2u)i+j=2.

1

2 Introduction

0.3 Basics of finite element method for linear problems

Standard second-order model problem (Poisson diffusion equation)

−∆u = f in Ω, u = 0 on ∂Ω, (0.3.1)

in a bounded domain Ω ⊂ Rd(d = 2ord = 3) with sufficiently regular boundary ∂Ω. Corre-

sponding fourth-order problem (plate bending problem)

∆2u = f in Ω, u = 0, ∂nu = 0 on ∂Ω. (0.3.2)

Variational formulation: Minimize the energy functional

E(u) := 12(∇u,∇u)− (f, u)

on the function Hilbert space (“energy space” consisting of functions with finite energy) V :=H1

0 (Ω). There exists a unique strict minimum, which is determined by the variational equation

(∇u,∇ϕ) = (f, ϕ) ∀ϕ ∈ V. (0.3.3)

We use the abstract formulation

a(u, ϕ) = l(ϕ) ∀ϕ ∈ V, E(u) = 12a(u, u) − l(u), (0.3.4)

with the “energy bilinear form” a(u, ϕ) := (∇u,∇ϕ) and the “load functional” l(ϕ) := (f, ϕ).For the plate bending problem there holds a similar formulation (exercise).

Finite element subspaces Vh ⊂ V parametrized by a mesh size parameter h > 0 . Discreteapproximations uh ∈ Vh are determined by the variational equations

a(uh, ϕh) = l(ϕh) ∀ϕh ∈ Vh. (0.3.5)

There holds “Galerkin orthogonality”

a(u− uh, ϕh) = 0, ϕh ∈ Vh, (0.3.6)

and consequently (for symmetric energy form) the “best approximation property”

‖u− uh‖E = minϕh∈Vh

‖u− ϕh‖E , (0.3.7)

with the “energy norm” ‖ · ‖E := a(·, ·)1/2. Notion of “Ritz projection” Rh : V → Vh ,

a(Ru, ϕh) = a(u, ϕh) ∀ϕh ∈ Vh. (0.3.8)

Improved convergence estimates in weaker norms by duality argument

‖u−Rhu‖ ≤ ‖u−Rhu‖E supz∈V ∩H2(Ω)

‖z −Rhz‖E . (0.3.9)

This is general theory for Ritz-Galerkin methods.

Special case of “finite element spaces” and approximation properties:

0.4 Useful technics of mathematical analysis in the finite element method 3

i) types of meshes:

hT := radius of minimal circumscribed ball (∼ diam(T ) ),

ρT := radius of maximal inscribed ball.

“structural regularity” (no hoes and ‘, “shape regularity”, “size regularity” - “quasi-uniformity”.“nodal interpolation” IT v ∈ P (T ) on cells T ∈ Th (“unisolvance property“), “nodal interpo-lation” Ihv ∈ Vh on meshes Th = T of sufficiently smooth functions (e. g., v ∈ C(Ω) .Interpolation error estimates for finite elements of order m ≥ 2 (polynomial degree m− 1 ≥ 1 ,special situation for m = 1 ):

‖∇(v − IT v)‖T ≤ cIhm−1T ‖∇mv‖T . (0.3.10)

Consequence, h := maxT∈ThhT ,

‖v − Ihv‖ =( ∑

T∈Th

‖v − IT v‖2T)1/2

≤ cI

( ∑

T∈Th

h2(m−1)T ‖∇mv‖2T

)1/2

≤ cIhm−1‖∇mv‖.

Other interpolation error estimates.

‖v − Ihv‖Ω ≤ chm‖∇mv‖Ω,‖v − Ihv‖∂Ω ≤ chm−1/2‖∇mv‖Ω,

‖v − Ihv‖L∞(Ω) ≤ c...

“Inverse relation” for finite elements:

‖∇vh‖T ≤ cρ−1T ‖vh‖T , T ∈ Th. (0.3.11)

Consequently, for quasi-uniform meshes:

‖∇vh‖Ω ≤ h−1‖vh‖Ω, vh ∈ Vh.

Questions to be considered in this context:

i) best-approximation estimates in Lp-norms (1 ≤ p ≤ ∞)?ii) relaxation of requirements on finite element meshes?

0.4 Useful technics of mathematical analysis in the finite element method

We present some technical arguments, which are frequently used in the error analysis for finiteelement approximations.

4 Introduction

0.4.1 Duality arguments (“Aubin-Nitsche trick”)

From the best-approximation property and the interpolation error estimates for finite elements,we obtain the basic energy-error estimate

‖∇(u− uh)‖ ≤ minϕh∈Vh

‖∇(u− ϕh)‖.

Auxiliary (“dual”) problem for e := u− uh ,

a(ϕ, z) =(e, ϕ)

‖e‖ ∀ϕ ∈ V.

Since Ω is assuned to be smoothly bounded or a convex polygonal domain the “dual solution”z ∈ V is in H2(Ω) and satisfies the a priori estimate

‖∇2z‖ ≤ ‖e‖ (constantc = 1).

Then, by Galerkin orthogonality,

‖e‖ = a(e, z) = a(e, z − ψh) ≤ ‖∇e‖minψh∈Vh‖∇(z −

0.4.2 Inverse estimates

There holds‖∇vh‖T ≤ cρ−1

T ‖vh‖TProof by transformation onto “reference unit cell” T → T General rule:

Variant in R2:

maxT

‖vh‖ ≤ ch−1‖vh‖T

0.4.3 Local integral inequalities

We estimate functions in H1(Ω) over lower dimensional faces Γ . This gives us the so-called“trace theorem” (∫

Γ|v|2 do

)1/2≤ c...

Estimate first proven for regular functions v ∈ C1(Ω) and then extended by continuity to itscompletion H1(Ω).

0.4.4 Lax-Milgram Lemma

The so-called Lax-Milgram Lemma is a generalization of the well-known Riesz representationtheorem for nonsymmetric bilinear forms. Let H be a (real) Hilbert space with scalar product(·, ·) and associated norm ‖ · ‖ = (·, ·)1/2 . Then, for any linear bounded (continuous) functionall(·) : H → R (i. e., any element from the “dual space” H∗ of H), there exists a unique elementv ∈ H such that

l(ϕ) = (v, ϕ), ϕ ∈ H,

0.5 Exercises (for refreshing the knowledge of some preparatory material) 5

and

‖v‖ = ‖l‖V ∗ := supϕ∈H

l(ϕ)

‖ϕ‖ .

The mapping l → v defines an isometric isomorphism between the Hilbert space H and itsdual H∗. In this sense the two spaces may be identified. The formally more general situationthat a(·, ·) is a bounded, symmetric bilinear form on H ,

a(u, v) = a(v, u), |a(u, v)| ≤ α‖u‖‖v‖, u, v ∈ H,

can be embedded into this situation if the bilinear form is positive definit,

a(v, v) ≥ γ‖v‖2, v ∈ H.

Then, the bilinear form a(·, ·) constitutes a scalar product on H , and its associated norm isequivalent to the given norm of H. The Lax-Milgram lemma addresses this situation for thecase that a(·, ·) is not symmetric, but bounded and positive definit.

0.5 Exercises (for refreshing the knowledge of some preparatory material)

Exercise 0.1: Give short answers to the following questions:

1. Which properties should an elliptic boundary value problem or a parabolic initial-boundaryvalue problem possess for being called “well-posed”?

2. Let the 1st BVP of the Laplace operator on the unit square be approximated by bilinearfinite elements on an equidistant cartesian quadrilateral mesh. Which finite differencescheme is obtained if the elements of the system matrix are computed using the tensor-product trapezoidal rule? What are the orders of the finite element scheme and of thisrelated finite difference scheme?

3. What are the dimensions of the polynomial spaces Q1(T ), P2(T ) and P5(T ) in R2 ?

4. On a tetrahedron T ∈ R3 let a polynomial space P (T ) and a set of functionals χr :

C1(Ω) → P (T ) (r = 1, . . . , R) be given. What does it mean that χrr=1,...,R is “unisol-vent” with respect to P (T ) ? What is the natural “nodal interpolation” Ihv ∈ P (T ) of acontinuous function v ?

5. What is the difference between the “Ritz projection method” and a general “Galerkinmethod”?

6. Explain the meaning of the terms “conformity”, “Galerkin orthogonality” and “best ap-proximation property” in the context of a finite element discretization.

7. Let the 1st BVP of the Laplace operator be discretized by a finite element method withnodal basis ϕih, i = 1, . . . , Nh . What are the corresponding “mass matrix” and “stiffnessmatrix”?

8. What is the h-dependence of the condition number of the system matrices of a finiteelement discretization for the Laplace operator and for the biharmonic operator on asequence of quasi-uniform meshes?

6 Introduction

9. What is the difference between the “method of lines” and the “Rothe method” for thediscretization of the parabolic heat equation? Why are the resulting linear ODE systemsin the “Method of lines” generically “stiff”?

Exercise 0.2: Give the best possible powers of h in the following error estimates for the nodalinterpolation IT v into the space of linear polynomials P (T ) := P1(T ) on a form-regular meshTh = T in 2d and a ∈ T (no proof required):

(i) ‖v − IT v‖T ≤ cih?T ‖∇2v‖T ,

(ii) |(v − IT v)(a)| ≤ cih?T ‖∇2v‖T ,

(iii) ‖∂n(v − IT v)‖∂T ≤ cih?T ‖∇2v‖T ,

for v ∈ H2(T ). Why is the estimate

‖v − IT v‖T ≤ cihT ‖∇v‖T

not possible uniformly for v ∈ H2(T )? State the corresponding best possible estimates for thenodal interpolation into the space of quadratic polynomials.

Exercise 0.3: Consider the Dirichlet BVP

−∆u+ γu = f in Ω, u = 0 on ∂Ω,

on a convex polygonal domain Ω ⊂ R2 . The data γ ≥ 0 and f are supposed to be sufficiently

regular. Let an approximate solution uh be computed by using the finite element method withsubspaces Vh ⊂ H1

0 (Ω) .

a) State the corresponding discrete and continuous variational formulations.

b) For this discretization with “linear” elements on a quasi-uniform family of triangulationsThh∈R+

state optimal-order a priori error estimates in the energy and the L2 norm. How doesthe condition number of the resulting system matrices Ah depend on the mesh width h ?

c) Consider part b), but now with “quadratic” elements.

Exercise 0.4: Give variational formulations in appropriate Sobolev spaces for the followingboundary value problems:

a) −∆u+ u = f in Ω, ∂nu = g on ∂Ω,

b) ∆2u = f in Ω, u = ∆u = 0 on ∂Ω,

c) −∇ · (a∇u) + β · ∇u = f in Ω, u = 0 on ∂Ω,

where the data a, β, f, and g are assumed to be sufficiently regular.

Exercise 0.5: Let β ∈ C1(Ω)2 be a transport vector function with the property ∇ · β = 0,and f ∈ L2(Ω). Prove by the Lax-Milgram lemma that the 2-d transport-diffusion problem

−∆u+ β · ∇u = f in Ω, u = 0 on ∂Ω,

possesses a unique weak solution in the Sobolev space V = H10 (Ω). Which regularity can be

expected for this solution if Ω is a convex polygonal domain?

0.5 Exercises (for refreshing the knowledge of some preparatory material) 7

Exercise 0.6: The standard “trace inequality”

‖v‖L2(∂Ω) ≤ c‖v‖H1(Ω)

is well known to hold for functions v ∈ H1(Ω) on domains Ω ⊂ Rd with sufficiently regular

boundary ∂Ω. Derive the following sharpened

version (for simplicity on the unit square Ω = (0, 1)2 ⊂ R2:

‖v‖L2(∂Ω) ≤ c‖v‖1/2H1(Ω)

‖v‖1/2L2(Ω)

,

(Hint: Modify the argument for proving the first “trace inequality” given in the lecture notes ofthe PDE Numerics course.)

Exercise 0.7: Consider the approximation of the Neumann problem

−∆u+ u = f in Ω, ∂nu = 0 on ∂Ω,

on a convex polygonal domain Ω ⊂ R2 by the finite element method using piecewise linear

finite elements on a quasi-uniform sequence of triangulations Th = T with mesh sizes h→ 0 .Prove the error estimates

‖u− uh‖L2(Ω) + h1/2‖u− uh‖L2(∂Ω) + h‖∇(u− uh)‖L2(Ω) ≤ ch2‖f‖L2(Ω).

(Hint: Use the standard arguments for proving error estimates for finite element approxima-tions such as “best approximation property”, optimal-order interpolation estimates, “dualityarguments”, and the trace estimate of Exercise 1.3.)

8 Introduction

1 Some Special Types of Nonlinear Problems

This chapter deals with the finite element solution of certain classes of nonlinear problems.Typical examples are the “obstacle problem” and the “minimal surface problem”. The materialof this chapter and further details can largely be found in the textbook of Ciarlet [26] and thearticle of Rannacher [45]. To get started, we recall the basic elliptic model problem

−∆u = f in Ω, u = g on ∂Ω, (1.0.1)

which possesses a unique “weak” solution in the linear manifold Vg := V + g, V := H10 (Ω) . For

this, we assume that the boundary value is given as the trace of a function g ∈ H1(Ω). Then, thisweak solution is characterized as minimizer on Vg of the quadratic energy functional, assumingthat f ∈ L2(Ω) ,

E(u) := 12‖∇u‖2 − (f, u),

or equivalently by the variational equation

a(u, ϕ) = l(ϕ) ∀ϕ ∈ V, (1.0.2)

with the notationa(u, ϕ) := (∇u,∇ϕ), l(ϕ) := (f, ϕ).

This is the mathematical model of an elastic membrane spanned over the horizontal domainΩ ⊂ R

2 and fixed along the boundary ∂Ω at values g undergoing a vertical deflection u(x)under a vertical load density f(x).

1.1 Examples of nonlinear problems

1.1.1 Minimization problems

The linear boundary value problem (1.0.1) has nonlinear extensions of different types:

a) The energy functional E(·) is minimized only over a convex subset V∗ of the function spaceV . A typical example is obtained for

V = H10 (Ω), V∗ = v ∈ V | v ≥ ψ a. e. in Ω,

where ψ is a prescribed obstacle function. This so-called “obstacle problem” describes thedeflection u of an elastic membrane spanned over a rigid obstacle ψ. This can also be writtenin a certain variational form as will be shown below.

b) The energy functional E(·) is not quadratic, e.g.,

i) Minimal surface problem: The membrane has the surface measure

F (u) =

∫

Ω

√1 + |∇u|2 dx.

9


The problem of determining the deflection u , which results in the minimal surface is called“minimal surface problem”.

min

∫

Ω

√1 + |∇u|2 dx on H1(Ω), v = g on ∂Ω. (1.1.3)

ii) “p-Laplace problem”:

min

∫

Ω

(1 + |∇u|2

)p/2dx−

∫

Ωfu dx on H1

0 (Ω). (1.1.4)

or, more general:

min

∫

ΩF (·, u.∇u) dx on H1

0 (Ω). (1.1.5)

If u ∈ H10 (Ω) is a minimizer of the functional E(·) , then there necessarily holds

d

dtE(·, u + tϕ,∇(u+ tϕ))

∣∣∣t=0

= 0 ∀ϕ ∈ H10 (Ω).

For the minimal surface problem this reads

∫

Ω

∇u · ∇ϕ√1 + |∇u|2

dx = 0, ∀ϕ ∈ H10 (Ω),

from which we get as necessary optimality condition the nonlinear boundary value problem(provided that the minimizer u is sufficiently regular)

−∇ ·( ∇u√

1 + |∇u|2)= 0 in Ω, u = g on ∂Ω.

This is called the “Euler equation” of the minimization problem. Analogously, we obtain for thegeneral energy functional the variational equation

∫

Ω

(F ′∇u(x, u,∇u) · ∇ϕ+ F ′

u(x, u,∇u)ϕ)dx = 0, ∀ϕ ∈ H1

0 (Ω),

and the corresponding Euler equation

−∇ · F ′∇u(x, u,∇u) + F ′

u(x, u,∇u) = 0 in Ω, u = g on ∂Ω.

This is called a quasi-linear second-order differential equation in “divergence form”.

1.1.2 Nonlinear diffusion-reaction-transport problems

The symmetric operator ∆ is supplemented by lower order nonlinear terms.

i) Reaction-diffusion problem (mathematical model in cell biology)

−D∆u = f(·, u), f(·, u) = u

1 + u2.

1.1 Examples of nonlinear problems 11

ii) Transport-diffusion problem−D∆u+ β(u) · u = f.

a) Example (nonstationary) Burgers equation in R1,

∂tu− ν∂2xu+ u∂xu = 0 on R1.

Exact solution (picture of graph of tanh(x))

u(x, t) = 1− tanh(x− t

2ν

), u(x, t) → 2 (x→ −∞), u(x, t) → 0 (x→ ∞).

Derivatives

dx tanh(x) = 1− tanh2(x),

∂xu(x, t) = − 1

2ν

(1− tanh2

(x− t

2ν

))= −∂tu(x, t).

ν∂2xu(x, t) = ν∂x(∂xu(x, t)

)

= −1

2∂x

(1− tanh2

(x− t

2ν

))= tanh

(x− t

2ν

)∂x tanh

(x− t

2ν

)

= − tanh(x− t

2ν

)∂x

(1− tanh

(x− t

2ν

))

=(1− tanh

(x− t

2ν

))∂x

(1− tanh

(x− t

2ν

))− ∂x

(1− tanh

(x− t

2ν

))

= u(x, t)∂xu(x, t) + ∂tu(x, t).

The functiona u(x) = 1− tanh(x2ν

)is solution of the stationary Burgers-type equation

−ν∂2xu+ (u− 1)∂xu = 0.

b) Example (stationary) Navier-Stokes equation in Rd (d = 2 or 3)

−ν∆v + v · ∇v +∇p = f, ∇ · v = 0 in Ω, v = 0 on ∂Ω.

v, p velocity vector and scalar pressure in an incompressible viscous Newtonian fluid. “No-slip”condition along the boundary of the flow domain. Examples “lid-driven cavity” and channelflow.

Exact solutions in special configurations:

i) Couette flow (parallel shear flow) Flow between two infinite plates (parallel to the (x1, x2)-plane with constant distance L = 1 , where the bottom one is kept fixed and the upper one ismoved with constant speed in x1-direction. The velocity vector

v1(x) = x3, v2(x) = v3(x) = 0 ,

is obviously divergence free and satisfies the no-slip condition at the plates. Together with thetrivial pressure p = 0 it satifies the Navier-Stokes equation.

−ν∆v + v · ∇v +∇p = 0 .


ii) Poiseuille flow (parabolic channel flow) Parallel flow through an infinite pipe parallel tothe x1-axis in 3d with circular cross section with radius R = 1. The velocity vector

v1(x) = 1− (x22 + x23), v2 = v3 = 0 ,

is divergence free and satisfies the np-slip condition along the wall of the pipe. We have

v · ∇v ≡ 0, ∆v1 = −4 ,

so that this velocity together with the linear pressure p(x) := −4νx1 satisfies the Navier-Stokesequation:

−ν∆v + v · ∇v +∇p = 0 ,

These two examples are among the very few for which solutions of the Navier-Stokes equationscan be written in closed form (without using series representations). They demonstrate thatin certain circumstances there are stationary solutions for arbitrary parameter values nu > 0 .Below, we will see that this is also the case in more general situations (e. g., domains).

1.1.3 Von Karman model in plate bending theory

In the case of “small” deflections of thin plates, |u| ≪ d≪ 1 , assuming linear material behavior(i. e., linear stress-strain relation) the governing model is the Kirchhoff plate equation , a linearfourth-order PDE. This reads with clamped boundary conditions:

∆2u =p

Din Ω, u = ∂nu = 0 on ∂Ω. (1.1.6)

The corresponding variational formulation uses the Sobolev space V = H20 (Ω) and the energy

forma(u, ϕ) = (∆u,∆ϕ) + (1− σ)(2∂1∂2u∂1∂2ϕ− ∂21u∂

22ϕ− ∂22u∂

21ϕ) dx,

with a materials parameter σ,. We note that that part of the nervy form which contains theparameter σ vanishes for functions in H2

0 (Ω). Therefore, the “weak” solution of the associatedvariational equation

a(u, ϕ) = (f, ϕ) ∀ϕ ∈ V.

corresponds to the plate problem (1.1.6), which is formally independent of the parameter σ.

In the description of thin plates with “large” (relative to the thickness < d ) deflection|u| ≈ d the deformation of the middle surface of the plate cannot be neglected anymore. Thecorresponding geometrically semi-linear theory goes back to von Karman (1910). It is an ex-tension of the linear Kirchhoff theory, which results in the following system of two fourth-orderequations:

∆2u =p

D+

1

D(∂22Ψ∂

21u+ ∂21Ψ∂

22u− 2∂1∂2Ψ∂1∂2u), (1.1.7)

∆2Ψ = Ed(∂1∂2u2 − ∂21u∂

22u), (1.1.8)

for the normal deflection u of the plate’s middle surface and for the so-called “stress function”Ψ , from which the horizontal distortion of the middle surface can be derived. The corresponding

1.2 Convex minimization problems and variational inequalities 13

boundary conditions are

u = ∂nu = 0 on ∂Ω, Ψ = ∂nΨ = 0 on ∂Ω. (1.1.9)

1.1.4 Eigenvalue problems

The eigenvalue problem of the Laplacian operator

−∆v = λv in Ω, v = 0 on ∂Ω, (1.1.10)

has the variational formulation in the space v = H10 (Omega) : Find a pair v, λ ∈ V ×R such

that

a(v, ϕ) = (∇v,∇ϕ) = λ(v, ϕ) ∀ϕ ∈ V. (1.1.11)

Because of the quadratic form λu this is a generically nonlinear problem. It corresponds to thenonlinear minimization problerm (Rayleigh quatient)

minv∈V

R(v) = λmin, R(v);=a(v, v)

‖v‖2 .

Alternatively it may also be written as nonlinear system of variational equations

a(v, ϕ) − λ(v, ϕ) = 0 ∀ϕ ∈ V,

(‖v‖2 − 1)χ = 0 ∀χ ∈ R.(1.1.12)

This last formulation also makes sense in the nonsymmetric case, formulated over the field C,and can be used as starting point of numerical approximations.

1.2 Convex minimization problems and variational inequalities

We consider the following abstract setting: Let be given a normed vector space V with norm‖ · ‖, a continuous (bounded) bilinear form a(·, ·) : V × V → R and a continuous (bounded)linear form l(·) : V → R ,

|a(v,w)| ≤ β‖v‖‖w‖, |l(v)| ≤ γ‖v‖, v, w ∈ V,

and a (nonempty) subset M ⊂ V . With the abstract “energy functional”

J(u) := 12a(u, u)− l(u),

we consider the minimization problem

min J(u) on M. (1.2.13)

Theorem 1.1: Under the following conditions the abstract minimization problem (1.2.13) pos-sesses a unique solution:(i) The normed vector space V is complete (i. e. a Banach space).(ii) The subset M ⊂ V is convex and closed.


(iii) The bilinear form a(·, ·) is symmetric and V -ellipic, i. e., there exists some constant α > 0such that

a(u, u) ≥ α‖u‖2, u ∈ V.

Proof 1.1: The bilinear form a(·, ·) defines a scalar product on V and the associated norm‖ · ‖a := a(·, ·)1/2 is equivalent to the norm ‖ · ‖ of V . Then, the space V equipped withthis scalar product is a complete inner product space, i. e. a Hilbert space. By the Rieszrepresentation theorem, there exists an element vl ∈ V such that

l(ϕ) = a(vl, ϕ), ϕ ∈ V.

Hence the functional J(·) can be rewritten in the form

J(v) = 12a(v, v) − l(v) = 1

2a(v, v) − a(vl, v) =12a(v − vl, v − vl)− 1

2a(vl, vl)

This shows that the minimization of J(·) on M is equivalent to minimizing the distance betweenthe element vf and the set M with respect to the norm ‖ · ‖a , i. e., the solution is simplythe projection of vl onto the set M with respect to the scalar product a(·, ·). By the “Hilbertprojection theorem” stated below such a projection exists and is uniquely determined. Q.E.D.

Lemma 1.1 (Hilbert projection theorem): Let V be a Hilbert space and M ⊂ V a closedconvex subset. Corresponding to any vector x ∈ V , there is a unique vector mx ∈M such that

‖x−mx‖ ≤ minm∈M

‖x−m‖. (1.2.14)

Furthermore, if M ⊂ V is a closed subspace a necessary and sufficient condition that mx ∈Mis the unique minimizing vector is that x−mx is orthogonal to M .

Proof 1.2: i) Existence of mx: Let δ ≥ 0 be the distance between x and M , and (xn)n∈N asequence in M such that

‖x− xn‖2 ≤ δ2 + 1/n, n ∈ N.

Let n,m ∈ N be arbitrary. Then,

‖xn − xm‖2 = ‖xn − x‖2 + ‖xm − x‖2 − 2(xn − x, xm − x)

and4‖1

2 (xn + xm)− x‖2 = ‖xn − x‖2 + ‖xm − x‖2 + 2(xn − x, xm − x).

This implies that

‖xn − xm‖2 = 2‖xn − x‖2 + 2‖xm − x‖2 − 4‖12 (xn + xm)− x‖2.

Consequently, using the definition of δ and the convexity of M ,

‖xn − xm‖2 ≤ 2(δ2 + 1/n) + 2(δ2 + 1/m)− 4δ2 = 2(1/n + 1/m).

This shows that the sequence (xn)n∈N is a Cauchy sequence in the closed subset M ∈ V andtherefore possesses a limit mx ∈M with minimal distance to x,


ii) Uniqueness of mx: Let m1x,m

2x be two minimizers. Then,

‖m2x −m1

x‖2 = 2‖m1x − x‖2 + 2‖m2

x − x‖2 − 4‖12 (m

1x +m2

x)− x‖2.

Since 12 (m

1x +m2

x) ∈M , we have

‖12 (m

1x +m2

x)− x‖2 ≥ δ2,

and therefore,‖m2

x −m1x‖2 ≤ 2δ2 + 2δ2 − 4δ2 = 0.

This means m1x = m2

x.iii) Orthogonality characterization: Let mx ∈M satisfy

(mx − x,m) = 0, m ∈M.

Then,

‖x−m‖2 = ‖mx − x‖2 + ‖m−mx‖2 + 2(mx − x,m−mx) = ‖mx − x‖2 + ‖m−mx‖2,

which shows that mx is a minimizer. On the other hand, for a minimizer mx , there holds withany m ∈M and t ∈ R that

‖(mx + tm)− x‖2 − ‖mx − x‖2 = 2t(mx − x,m) + t2‖m‖2 = 2t(mx − x,m) +O(t2) ≥ 0.

This implies that necessarily (mx − x,m) = 0 .

Theorem 1.2: a) An element u ∈M ⊂ V is the solution of the abstract minimization problem(1.2.13) if ans only if it satisfies the variational inequality

a(u, v − u) ≥ l(v − u) ∀v ∈M. (1.2.15)

b) In case that M is a closed subspace the necessary and sufficient relation becomes a variationalequality,

a(u, v) = l(v) ∀v ∈M. (1.2.16)

Proof 1.3: a) Let u ∈M be the minimizer of (1.2.13). Since M is convex, for any v ∈M allelements of the form u+ t(v − u) = tv + (1− t)u, t ∈ [0, 1], are in M . Then, the continuouslydifferentiable function

j(t) := J(u+ t(v − u)), t ∈ [0, 1],

has a minimum at t = 0 , and therefore,

d

dtj(t)

∣∣∣t=0

=d

dt

(a(u+ t(v − u), u+ t(v − u))− l(u+ t(v − u))

)∣∣∣t=0

≥ 0.

This implies the asserted variational inequality

a(u, v − u) ≥ l(v − u) ∀v ∈M.

Let now this variational inequality be satisfied by some u ∈ M . Then, for any v ∈ M there


holds

0 ≤ a(u, v − u)− l(v − u)

= −a(u, u) + l(u) + a(u, v) − l(v)

= −12a(u, u) + l(u)− 1

2a(u, u) + a(u, v) − 12a(v, v) − l(v) + 1

2a(v, v)

= −J(u)− 12‖u− v‖2a + J(v),

and thusJ(u) + 1

2‖u− v‖2a ≤ J(v).

This shows that u is a strict minimizer of J(·) on M .

b) We use the inequality (1.2.15) with u+ v ∈M for v,−v ∈M to obtain

a(u, v) ≥ l(v), a(u, v) ≤ l(v) ∀v ∈M.

This implies the asserted variational equation (1.2.16).

Remark 1.1: In case that the subset M ⊂ V is a closed convex cone with vertex at the origin,i. e., u, v ∈ M implies αu + βv ∈ M for α, β ∈ R+ , then necessary and sufficient variationalinequality for the minimizer u ∈M takes the form

a(u, v) ≥ l(v) ∀v ∈M, a(u, u) = f(u). (1.2.17)

This is an intermediate result of the above proof of the variational equation (1.2.16).

Remark 1.2: If the bilinear form a(·, ·) is nonsymmetric, we may consider the variationalinequality

a(u, v − u) ≥ l(v − u) ∀v ∈M,

though, in this case, it is not related to a minimization problem. The result of Theorem 1.1can be extended to this situation (if V itself is a Hilbert space) in such that the variationalinequality has a unique solution in M . In the case that M is a linear subspace or even morespecial that M = V this is just the content of the well-known “Lax-Milgram lemma”.

1.2.1 Approximation of abstract variational inequalities

First, we develop an error estimate for the above abstract variational inequality, from whichafterwards concrete error estimates for the approximation of obstacle problems will be derived.

We consider the following somewhat more special setting: Let V be a Hilbert space withnorm ‖ · ‖V , a(·, ·) a bounded, symmetric, V -elliptic bilinear form and f(·) a bounded linearform on V . Then, for any closed convex subset M ⊂ V there exists unique minimizer u ∈Mof the abstract energy functional J(·) := 1

2a(·, ·) − f(·) on M , which is characterized by thevariational inequality

a(u, v − u) ≥ f(v − u) ∀v ∈M. (1.2.18)

For discretizing this problem, we consider a finite dimensional subspace Vh ⊂ V and a (nonempty) closed, convex subset Mh ⊂ Vh not necessarily contained in M . The approximation


uh ∈Mh to u is then defined as the unique solution of the “discrete” variational inequality

a(uh, vh − uh) ≥ f(vh − uh) ∀vh ∈Mh. (1.2.19)

The bilinear form a(·, ·) defines an operator A : V → V ∗ through the relation

Au(ϕ) := a(u, ϕ), ϕ ∈ V.

In this sense Au − f may be considered as a functional in the dual space V ∗ . Notice that inthe present situation, we generally do not have Au = f .

Theorem 1.3: For the approximation of the abstract variational inequality, there holds

‖u− uh‖V ≤ c(

infvh∈Mh

‖u− vh‖2V + |(Au− f)(u− vh)|

+ infv∈M

|(Au− f)(uh − v)|)1/2

,

(1.2.20)

with a constant c independent of the choice of Vh and Mh.

Proof: We have

α‖u − uh‖2V ≤ a(u− uh, u− uh) = a(u, u) + a(uh, uh)− a(u, uh)− a(uh, u),

and by the variational inequalities satisfied by u and uh :

a(u, u) ≤ a(u, v) + f(u− v), v ∈M,

a(uh, uh) ≤ a(uh, vh) + f(uh − vh), vh ∈Mh.

Therefore, we conclude that for all v ∈M and vh ∈Mh :

α‖u− uh‖2V ≤ a(u, v − uh) + a(uh, vh − u) + f(u− v) + f(uh − vh)

= a(u, v − uh)− f(v − uh) + a(u, vh − u)− f(vh − u) + a(uh − u, vh − u)

= (f −Au, uh − v) + (f −Au, u− vh) + a(u− uh, u− vh),

and, consequently,

α‖u− uh‖2V ≤ |(f −Au)(u− vh)|+ |(f −Au)(uh − v)|+ β‖u− uh‖V ‖u− vh‖V .

Since

‖u− uh‖V ‖u− vh‖V ≤ 1

2

αβ‖u− uh‖2V +

β

α‖u− vh‖2V

,

we obtain, by combining the two previous inequalities,

α

2‖u− uh‖2V ≤ |(f −Au)(u− vh)|+ |(f −Au)(uh − v)|+ β2

2α‖u− vh‖2V .

From this, we conclude the asserted estimate. Q.E.D.

Remark 1.3: (i) The above proof also covers the case of a nonsymmetric bilinear form a(·, ·).


(ii) If the inclusion Mh ⊂M holds, then the difficult term infv∈M |(Au−f)(uh−v)|1/2 vanishes.

(iii) In the case M = V , then Au = f and (1.2.20) reduces to the usual best-approximationextimate of the linear situation.

1.2.2 Application to obstacle and Signorini problem

Obstacle problem

We consider the obstacle problem on a convex polygonal domain Ω ⊂ R2 , with the setting

V = H10 (Ω), M = v ∈ V, v ≥ ψ a. e. on Ω, H = L2(Ω),

with norms ‖v‖V := ‖∇v‖L2 = ‖v‖1, ‖v‖H := ‖v‖L2 = ‖v‖ , and

a(u, ϕ) = (∇u,∇ϕ), f(ϕ) = (f, ϕ),

with a function f ∈ L2(Ω) and a smooth obstacle function ψ ∈ H10 (Ω) ∪ H2(Ω). Further,

let Vh|, be the usual spaces of piecewise linear finite elements on a quasi-uniform sequence oftriangulations Thh>0 . We choose the discrete set

Mh := vh ∈ Vh | vh ≥ ψh,

where Ihψ is the usual piecewise linear nodal interpolation of ψ , which is well defined asψ ∈ H2(Ω). Generally, Mh is not contained in M

Corollary 1.1 (Obstacle problem): Suppose for the minimizer of the obstacle problemthatu ∈ H2(Ω). Then, for quasi-uniform sequence of triangulations there holds the error estimate

‖u− uh‖1 ≤ c(f, u, ψ)h, (1.2.21)

with a constant c(f, u, ψ) independent of h.

Proof: By assumption, we have f ∈ H = L2(Ω) and u ∈ H2(Ω). Therefore,

Au(v) := (∇u,∇v) = −(∆u, v), v ∈ V = H10 (Ω),

and thus|(Au)(v)| ≤ ‖∆u‖‖v‖.

This implies that indeed Au ∈ H . Further, by construction Ihu ∈Mh and therefore

infvh∈Mh

‖u− vh‖21 + ‖Au− f‖‖u− vh‖

≤ ‖u− Ihu‖21 + ‖Au− f‖‖u− Ihu‖

≤ ch2‖u‖22 + c‖Au− f‖h2‖u‖2.

It remains to evaluate the term infv∈M ‖uh − v‖ . To this end, we introduce the functionu∗h := maxuh, ψ , which satisfies u∗h ≥ ψ on Ω. Since both uh, ψ ∈ H1

0 (Ω) also u∗h ∈ H10 (Ω)

(nontrivial result) and then by construction also u∗h ∈M . With the set Λh := x ∈ Ω, uh ≤ ψ


observing uh − u∗h = 0 on Ω \ Γh, there holds,

‖uh − u∗h‖2 =∫

Λh

|uh − ψ|2 dx.

By construction, we have uh − Ihψ ≥ 0 and consequently on Λh:

0 < |ψ − uh| = ψ − uh ≤ ψ − Ihψ = |ψ − Ihψ|.

Thus,

‖uh − u∗h|2 ≤∫

Λh

|ψ − Ihψ|2 dx ≤ ‖ψ − Ihψ‖2 ≤ ch4‖ψ‖22,

and finally,infv∈M

‖uh − v‖ ≤ ch2‖ψ‖2,

which concludes the proof. Q.E.D.

Signorini problem

We consider a special Signorini problem again on a convex polygonal domain Ω ⊂ R2 , with the

settingV = H1(Ω), M = v ∈ V, v ≥ 0 a. e. on ∂Ω, H = L2(Ω),

with norms ‖v‖V := ‖v‖1 =(‖v‖2 + ‖∇v‖2

)1/2, ‖v‖H := ‖v‖L2 = ‖v‖ , and

a(u, ϕ) = (∇u,∇ϕ) + (u, ϕ), f(ϕ) = (f, ϕ),

with a function f ∈ L2(Ω) . Further, let Vh be again the usual spaces of piecewise linear finiteelements on a quasi-uniform sequence of triangulations Thh>0 . We choose the discrete set

Mh := vh ∈ Vh | vh ≥ 0 on ∂Ω,

which this time is contained in M .

Corollary 1.2 (Signorini problem): Suppose for the minimizer of the Signorini problem thatu ∈ H2(Ω). Then, for quasi-uniform sequence of triangulations there holds the error estimate

‖u− uh‖1 ≤ c(f, u)h3/4, (1.2.22)

with a constant c(f, u) independent of h.

Proof: In this special situation the abstract error estimate (1.2.20) reduces to

‖u− uh‖1 ≤ c(

infvh∈Mh

‖u− vh‖21 + |(Au− f)(u− vh)|

)1/2.

Again by assumption, we have f ∈ H = L2(Ω) and u ∈ H2(Ω). Therefore,

Au(v) := (∇u,∇v) + (u, v) = −(∆u, v) + (∂nu, v)∂Ω + (u, v), v ∈ V = H1(Ω),


and thus, employing the trace inequality ‖∂nu‖∂Ω ≤ c‖u‖2 ,

|Au(v)| ≤ c‖u‖2(‖v‖ + ‖v‖∂Ω).

Further, by construction Ihu ∈Mh and therefore

infvh∈Mh

‖u− vh‖21 + |(Au− f)(u− vh)|

≤ ‖u− Ihu‖21 + c‖u‖2‖u− Ihu‖∂Ω

≤ ch2‖u‖22 + c‖u‖2h3/2‖u‖2,

which completes the proof. Q.E.D.

1.3 The minimal surface problem

Let Ω ⊂ R2 be a convex polygonal domain. We consider the minimization of the functional

J(u) :=

∫

Ω

√1 + |∇u|2 dx

on the manifold Vg := v ∈ V := H1(Ω) | v = g on ∂Ω for a prescribed function g ∈ H2(Ω) .This functional represents the surface content of the graph of the function u = u(x), x ∈ Ω ,what suggests the name “minimal surface problem”. In contrast to the obstacle problems thesolvability analysis of this problem is difficult and not discussed here. For the following, werather assume the existance of a minimizer in the manifold Vg .

Above, we have already seen that any solution u ∈ Vg of the minimal surface problemnecessarily satisfies the variational equation

∫

Ω

∇u · ∇ϕ√1 + |∇u|2

dx = 0 ∀ϕ ∈ V0 := H10 (Ω). (1.3.23)

This result can be expressed in the language of the Calculus of Variations in function spaces asfollows: The gradient and the Hessian matrix of the function

f(x) :=√

1 + |x|2, x ∈ R2,

act as follows:

(f ′(x), ξ) =

2∑

i=1

∂if(x)xiξi =x · ξ

(1 + |x|2)1/2 ,

(f ′′(x)ξ, ξ) =

2∑

i,j=1

∂i∂jf(x)ξiξj =|ξ|2 + (x2ξ1 − x1ξ2)

2

(1 + |x|2)3/2 .

Therefore, we see that

|ξ|2(1 + |x|2)3/2 ≤ (f ′′(x)ξ, ξ) ≤ |ξ|2, ξ ∈ R

2,

and particularly that f(·) is strictly convex on bounded sets. With this notation by Taylor

1.3 The minimal surface problem 21

expansion,

J(v + ϕ)− J(v) =

∫

Ω

√1 + |∇(v + ϕ)| −

√1 + |∇v|

dx

=

∫

Ω

∇v · ∇ϕ√1 + |∇v|2

dx+R(v, ϕ),

with a bounded remainder term

|R(v, ϕ)| ≤ 1

2

∫

Ω‖∇ϕ‖2 dx ≤ 1

2‖ϕ‖21.

Thus, the functional J(·) is differentiable on V with so-called “Frechet derivative” J ′(·) actingas a bounded functional on V like

J ′(v)ϕ =

∫

Ω

∇v · ∇ϕ√1 + |∇v|2

dx.

In this framework the above necessary optimality condition for the minimizer of the functionalJ(·) can be written in the form

J ′(u)ϕ = 0, ϕ ∈ H10 (Ω). (1.3.24)

Furthermore, there holds

J(u+ ϕ)− J(u) ≥∫

Ω

|∇ϕ|2(1 + |∇u|2)3/2 dx,

so that any minimizer u ∈ V + g (if it exists) is unique.

1.3.1 Finite element approximation

Let Thh>0 be a quasi-uniform family of triangulations of the (for simplicity) polygonal domainΩ and Vh ⊂ H1(Ω) the corresponding spaces of piecewise linear finite elements. With thenatural nodal interpolation gh := Ihg ∈ Vh of g (well defined since g ∈ H2(Ω) by assumption),we introduce the manifolds

Vh,gh := vh ∈ Vh | vh − gh = 0 on ∂Ω.

Then, the discrete approximations uh are defined as minimizers of the functional J(·) on Vh,gh .For later purposes, we also introduce the discrete spaces Vh,0 := Vh ∩ V0 .

Theorem 1.4 (Discrete minimal surface problem): There are uniquely determined mini-mizers uh ∈ Vh of the functional J(·) on the manifolds Vh,gh .

Proof: Let v∗h ∈ Vh,gh be an arbitrary but fixed function. Then, since on Vh all norms areequivalent, there exists some R > 0 such that

J(v∗h) ≤ J(vh), vh ∈ Vh,gh , ‖vh‖H1 ≥ R.


Hence, minimizing J(·) on Vh,gh is equivalent to minimizing it on the bounded subset vh ∈Vh,gh | ‖vh‖H1 ≤ R . Since this set is compact, the continuous functional J(·) has there aminimum. This minimum is also unique since the functional J(·) is strictly convex on boundedsets. Q.E.D.

Theorem 1.5: Suppose that the solution u ∈ Vg of the minimal surface problem exists and isin H2(Ω) ∩W 1,∞(Ω) . Then there holds the error estimate

‖u− uh‖H1 ≤ c(u)h, (1.3.25)

with a constant c(u) independent of h.

Proof: (i) As consequences of the minimization property of the functions u ∈ Vg and uh ∈Vh,gh , we have the relations

J ′(u)ϕ = 0, ϕ ∈ V0,

J ′(uh)ϕh = 0, ϕh ∈ Vh,0.

(ii) Next, we consider the quantity

∆h :=( ∫

Ω

|∇(u− uh)|2√1 + |∇uh|2

dx)1/2

,

and want to prove that ∆h ≤ c(u)h . Let vh ∈ Vh,gh be arbitrary, so that wh := vh−uh ∈ Vh,0 .Then, again setting f(x) :=

√1 + |x|2 and observing J ′(u)wh = 0 ,

∆2h =

∫

Ω

|∇(u− uh)|2√1 + |∇uh|2

dx

=

∫

Ω

∇(u− uh) · ∇(u− vh)

f(∇uh)dx+

∫

Ω

∇(u− uh) · ∇whf(∇uh)

dx

=

∫

Ω

∇(u− uh) · ∇(u− vh)

f(∇uh)dx+

∫

Ω

∇u · ∇whf(∇uh)

dx

=

∫

Ω

∇(u− uh) · ∇(u− vh)

f(∇uh)dx+

∫

Ω

( 1

f(∇uh)− 1

f(∇u))∇u · ∇wh dx .

For the first integral on the right, observing that f(x) ≥ 1 , we have

∣∣∣∫

Ω

∇(u− uh) · ∇(u− vh)

f(∇uh)dx

∣∣∣ ≤∫

Ω

|∇(u− uh)|√f(∇uh)

|∇(u− vh)| dx ≤ ∆h‖∇(u− vh)‖.

1.3 The minimal surface problem 23

In order to estimate the second integral on the right, we note that

1

f(∇uh)− 1

f(∇u) =f(∇u)− f(∇uh)f(∇uh)f(∇u)

=f(∇u)2 − f(∇uh)2

f(∇uh)f(∇u)(f(∇uh) + f(∇u))

=|∇u|2 − |∇uh|2

f(∇uh)f(∇u)(f(∇uh) + f(∇u))

=∇(u− uh)

f(∇uh)f(∇u)· ∇(u+ uh)

f(∇uh) + f(∇u) ,

and thus ∣∣∣ 1

f(∇uh)− 1

f(∇u)∣∣∣ ≤ |∇(u− uh)|

f(∇uh)f(∇u).

This is used to estimate

∣∣∣∫

Ω

( 1

f(∇uh)− 1

f(∇u))∇u · ∇wh dx

∣∣∣ ≤∫

Ω

|∇u|f(∇u)

|∇(u− uh)|√f(∇uh)

|∇wh|√f(∇uh)

dx

≤ γ(u)∆h

( ∫

Ω

|∇wh|2f(∇uh)

dx)1/2

≤ γ(u)∆h

(∆h + ‖∇(u− vh)‖

),

where, since by assumption u ∈W 1,∞(Ω) (crucial at this point),

γ(u) := ess supΩ|∇u|f(∇u) < 1.

Combining the foregoing estimates, we obtain

∆h ≤ γ(u)∆h + (1 + γ(u))‖∇(u − vh)‖).

Since γ(u) < 1 , it follows that

∆h ≤ c(u) infvh∈Vh+Ihg

‖∇(u− vh)‖,

where c(u) := (1 + γ(u))((1 − γ(u)) . Since by assumption u ∈ H2(Ω) the nodal interpolationIhu ∈ Vh is well defined and by construction satisfies Ihu ∈ Vh,gh. Thus,

infvh∈Vh,gh

‖∇(u− vh)‖ ≤ ch‖u‖2,

and therefore, ∆h ≤ c(u)h .

(iii) Next, we want to show that supΩ |∇uh| ≤ c(u) . Let T ∈ Th be an arbitrary cell. By thetriangle inequality and the result of (ii),

( ∫

T

|∇uh|2√1 + |∇uh|2

dx)1/2

≤ ∆h +(∫

T

|∇u|2√1 + |∇uh|2

dx)1/2

≤ c(u)h + ess supT |∇u|meas(T )1/2 ≤ c(u)h,


Since ∇uh|T is constant, we can write

∫

T

|∇uh|2√1 + |∇uh|2

dx =|∇uh|T |2√1 + |∇uh|T |2

meas(T ) ≥ c|∇uh|T |2√1 + |∇uh|T |2

h2.

The foregoing estimates imply that necessarily

maxT∈Th

|∇uh|T |2√1 + |∇uh|T |2

≤ c(u).

Observing x2/√1 + x2 → ∞ for x→ ∞ , we conclude the desired bound

supΩ

|∇uh| ≤ c(u).

(iv) Finally, combining the foregoing results, we estimate as follows:

‖∇(u− uh)‖ =( ∫

Ω

|∇(u− uh)|2√1 + |∇uh|2

√1 + |∇uh|2

)1/2

≤ ∆h

(maxT∈Th

√1 + |∇uh|T |2

)1/2≤ c(u)h.

(v) For estimating the full H1-norm of the error, we use the well-known inequality

‖v‖Ω ≤ c(‖∇v‖Ω + ‖v‖∂Ω

), v ∈ H1(Ω),

which, as generalization of the Poincare inequality, can be proven by a similar argument as hasbeen used in deriving the trace inequality. Employing this inequality for u− uh and observingthat

‖u− uh‖∂Ω = ‖g − Ihg‖∂Ω ≤ c(u)h3/2,

we obtain the asserted estimate ‖u− uh‖H1 ≤ c(u)h . Q.E.D.

Remark 1.4: By a much more involved argument one can prove the optimal-order L2-errorestimate

‖u− uh‖ ≤ c(u)h2. (1.3.26)

1.4 Problems of monotone type

In this section, we consider convex minimization problems of the form

min J(v) on V, J(v) :=1

p

∫

Ω|∇v|p dx−

∫

ωfv dx

, (1.4.27)

or, as already stated above,

minJ(v) on V, J(v);=1

p

∫

Ω

(1 + |∇v|2

)p/2dx−

∫

ωfv dx

, (1.4.28)

1.4 Problems of monotone type 25

for some p ∈ (1,∞) and an appropriate function space V . For p = 2 this corresponds to theusual quadatic minimization problem associated with the Laplacian operator. The limit casep = 1 is related to the minimal surface problem, which has turned out to be particularly difficultfor theoretical analysis. Here, we will restrict ourselves to the case p ≥ 2 and the first examplecorresponding to the so-called “p-Laplacian operator”. The case p < 2 and the other examplecan be treated with slightly different arguments leading to similar results.

The natural solution space for this minimization problems corresponding to homogeneousboundary values is the Sobolev (Banach) space V := H1,p

0 (Ω) , which may be defined as theclosure of the space C∞

0 (Ω) of test functions with respect to the norm

‖v‖1,p :=(‖v‖p0,p + ‖v‖p0,p

)1/p,

where ‖·‖p = ‖·‖0,p denotes the usual Lp-norm over the domain Ω . As usual, for simplifying theanalysis and the finite element approximation, the domain Ω is assumed to be two-dimensional,polygonal and convex. Further, the force term is assumed to be f ∈ L2(Ω) . The space Vis equipped with the norm ‖v‖V := ‖∇v‖p , which inview of the Lp-version of the Poincareinequality

‖v‖p ≤ c(p,Ω)‖∇v‖p, v ∈ H1,p(Ω), 1 < p <∞,

is equivalent to the norm ‖ · ‖1,p .

Remark 1.5: The dual V ∗ (space of all linear continuous functionals) of the space V =H1,p

0 (Ω) denoted by H−1,q(Ω) is likewise a Banach space. For 1 < p < ∞ the bi-dual (V ∗)∗

has a natural linear embedding into V , which is an isomophism, i. e., V is “reflexive”. Inreflexive Banach spaces any bounded sequence is “weakly compact”, i. e., contains a weaklyconvergent subsequence, a property which will be used below in proving existence of minimizersfor the functional J(·) . The difficulty with the minimal surface problem is partially due to thefact that for p = 1 the corresponding Banach space H1,1

0 (Ω) is not reflexive.

As argued before, any minimizer of J(·) on V necessarily satisfies the variational equation(exercise)

∫

Ω|∇u|p−2∇u · ∇ϕdx =

∫

Ωfϕdx ∀ϕ ∈ V. (1.4.29)

We want to embed this problem into a more abstract functional analytic setting. To this end, wenote that the left-hand side of the equation (1.4.29) defines a (nonlinear) operator A : V → V ∗

by

Au(ϕ) :=

∫

Ω|∇u|p−2∇u · ∇ϕdx, ϕ ∈ V.

For seeing this, we use the general Holder inequality (exercise)

∣∣∫

Ωvw dx

∣∣∣ ≤( ∫

Ω|v|p dx

)1/p(∫

Ω|w|q dx

)1/q, v ∈ Lp(Ω), w ∈ Lq(Ω), 1/p + 1/q = 1.

This is the same concept as that already used in the case of a linear bilinear form a(·, ·) leading toa likewise linear operator A : V → V ∗ . The following lemma provides natural generalizations ofthe properties “V -ellipticity” and “boundedness” (resp. “Lipschitz continuity”) for the nonlinearoperator A .


Lemma 1.2: For the operator A : V → V ∗ there hold the following inequalities:

α‖u− v‖pV ≤ (Au−Av)(u − v), u.v ∈ V, (1.4.30)

‖Au−Av‖V ∗ ≤ β(‖u‖V + ‖v‖V

)p−2‖u− v‖V , u.v ∈ V, (1.4.31)

with some constants α, β > 0 .

Proof: (i) We introduce the auxiliary function

ϕ(ξ, η) :=(|ξ|p−2ξ − |η|p−2η) · (ξ − η)

|ξ − η|p , ξ, η ∈ R2, ξ 6= η.

We want to show that there is some constant α > 0 , such that

α ≤ ϕ(ξ, η), ξ, η ∈ R2, ξ 6= η. (1.4.32)

From this, we obtainα|ξ − η|p ≤ (|ξ|p−2ξ − |η|p−2η) · (ξ − η),

and then by integration over Ω the estimate (1.4.30). Since

ϕ(0, η) = 1 for η 6= 0,

it suffices to consider the case ξ 6= 0 . Next, we prove that

ϕ(ξ, η) > 0 for ξ 6= η,

This follows from the relations

(|ξ|p−2ξ − |η|p−2η) · (ξ − η) = |ξ|p − (|ξ|p−2 + |η|p−2)(ξ · η) + |η|p

≥ |ξ|p − |ξ|p−1|η| − |η|p−1|ξ|+ |η|p

= (|ξ|p−1 − |η|p−1)(|ξ| − |η|)> 0 for |ξ| 6= |η|.

Since the penultimate inequality is an equality if and only if η = µξ for some µ ∈ R , the onlyremaining case is that where η = −ξ , But then

(|ξ|p−2ξ − |η|p−2η) · (ξ − η) = 4|ξ|p > 0.

We may restrict ourselves without loss of generality to the case ξ = ξ = (1, 0) since ϕ(λξ, λη) =ϕ(ξ, η) for all λ > 0 and since the Euclidean scalar product is invariant under rotations aroundthe origin. In view of

lim|η|→∞

ϕ(ξ, η) = 1,

it remains to to study the behavior of the function ϕ(ξ, η) in the neighborhood of the point ξ .To this end, let

η1 = 1 + ρ cos θ, η2 = ρ sin θ.


Then, a simple computation shows that

ϕ(ξ, η) =1 + (p − 2) cos2 θ + ε(ρ, θ)

ρp−2,

with limρ→0 ε(ρ, θ) = 0 uniformly for θ ∈ [9, 2π) . Therefore,

limη→ξ

ϕ(ξ, η) = 1 for p = 2, limη→ξ

ϕ(ξ, η) = ∞ for p > 2,

and the desired inequality (1.4.32) follows from the foregoing results.

(ii) To prove the inequality (1.4.31), we introduce the auxiliary function

ψ(ξ, η) :=

∣∣|η|p−2η − |ξ|p−2ξ∣∣

|η − ξ|(|η| + |ξ|)p−2, ξ, η ∈ R

2, ξ 6= η.

We want to show that there is some constant β > 0 such that

ψ(ξ, η) ≤ β, ξ, η ∈ R2, ξ 6= η. (1.4.33)

Sinceψ(0, η) = 1 for η 6= 0,

we can without loss of generality assume that ξ 6= 0 . Further, it suffices to consider the caseξ = ξ = (1, 0) , since ψ(λξ, λη) = ψ(ξ, η) for λ > 0 and since the Euclidian norm is invariantunder rotations around the origin. There also holds

lim|η|→∞

ψ(ξ, η) = 1.

To study the behavior of the function ψ(ξ, η) in the neighborhood of of the point ξ , we letagain

η1 = 1 + ρ cos θ, η2 = ρ sin θ,

For this, we obtain

ψ(ξ, η) = 22−p(1 + p(p− 2) cos2 θ

)1/2+ ε(ρ, θ),

with limρ→0 ε(ρ.θ) = 0 uniformly for θ ∈ [0, 2π) and therefore,

lim supη→ξ ψ(ξ.η) <∞.

These relations finally imply the desired estimate (1.4.33). As a consequence, we have

∣∣∣∣|η|p−2η − |ξ|p−2ξ∣∣ ≤ β|ξ − η|(|ξ| + |η|), ξ, η ∈ R

2. (1.4.34)

To prove (1.4.31), we use the characterization

‖Au−Av‖V ∗ = supw∈V

|(Au−Av)(w)|‖w‖V

. (1.4.35)


From the last inequality above, we conclude that

|(Au−Av)(w)| =∣∣∣∫

Ω(|∇u|p−2∇u− |∇v|p−2∇v) · ∇w dx

∣∣∣

≤∫

Ω

∣∣|∇u|p−2∇u− |∇v|p−2∇v∣∣ ]∇w| dx

≤ β

∫

Ω|∇(u− v)|(|∇u| + |∇v|)p−2|∇w| dx

≤(∫

Ω|∇(u− v)|p dx

)1/p(∫

Ω(|∇u|+ |∇v|)p dx

)(p−2)/p(∫

Ω|∇w|p dx

)1/p

≤ β‖u− v‖V (‖u‖V + ‖v‖V )p−2‖w‖V ,

where the generalized Holder inequality has been used with the exponents 1/p+(p−2)/p+1/p =1 . In view of the characterization (1.4.35) this completes the proof of inequality (1.4.31). Q.E.D.

1.4.1 An abstract error analysis

Motivated by the concrete problem presented above, we now consider the following abstractsetting. Let V be a reflexive Banach space with norm ‖ · ‖V and corresponding dual space V ∗

with norm ‖ ·‖V ∗ . Further let be given a (generally nonlinear) operator A : V → V ∗ , such thatA0 = 0 (for simplicity), and an element f ∈ V ∗ for which the equation

Au(ϕ) = f(ϕ) ∀ϕ ∈ V, (1.4.36)

is to be solved. For discretizing problem (1.4.36), we consider finite dimensional subspacesVh ⊂ V and the corresponding discrete problems

Auh(ϕh) = f(ϕh) ∀ϕh ∈ Vh. (1.4.37)

Here, the existence of solutions u ∈ V and uh ∈ Vh to these problems is assumed. Below,we will prove this to be actually true for the special case of the p-Laplace problem. We wantto derive an error estimate for the present abstract setting. To this end, we pose the followingconditions:

i) The mapping A : V → V ∗ is “strongly monotone”. i. e., there exists a strictly increasingfunction χ : [0,∞) → R with the properties χ(0) = 0 and χ(t) → ∞ (t → ∞) , such that

(Av −Aw)(v − w) ≥ χ(‖v − w‖V )‖v − w‖V , v, w ∈ V. (1.4.38)

In view of Lemma 1.2 the operator A corresponding to the p-Laplacian is strongly monotonewith χ(t) = αtp−1 .

ii) The mapping A : V → V ∗ is “Lipschitz continuous (for bounded arguments)”, i. e., for anyball BR(0) = v ∈ V | ‖v‖V ≤ R there exist a constant Γ(R) , such that

‖Av −Aw‖V ∗ ≤ Γ(R)‖v − w‖V , v, w ∈ BR. (1.4.39)

In view of Lemma 1.2 the operator A corresponding to the p-Laplacian is Lipschitz continuousfor bounded arguments with Γ(R) = β(2R)p−2 .


Theorem 1.6 (Abstract error estimate): Under the above assumptions there holds the er-ror estimate

χ(‖u− uh‖V ) ≤ c infvh∈Vh

‖u− vh‖V , (1.4.40)

with a constant c independent of the choice of Vh .

Proof: From the monotonicity and Lipschitz-continuity of A , we conclude that

χ(‖u‖V )‖u‖V = χ(‖u− 0‖V )‖u− 0‖V ≤ (Au−A0)(u − 0)

= f(u) ≤ ‖f‖V ∗‖u‖V ,

and consequently,χ(‖u‖V ) ≤ ‖f‖V ∗ .

In the same way, we obtain a corresponding estimate for uh . The assumed properties of thefunction χ(·) imply that it is invertible, so that

‖u‖V , ‖uh‖V ≤ χ−1(‖f‖V ∗).

Next, we fix an arbitrary element vh ∈ Vh . Using the equations satisfied by u and uh , weobtain

(Au−Auh)(ϕh) = 0, ϕh ∈ Vh,

and, in particular,(Au−Auh)(uh − vh) = 0.

Hence, combining the foregoing results,

χ(‖u− uh‖V )‖u− uh‖V ≤ (Au−Auh)(u− uh) = (Au−Auh)(u− vh)

≤ ‖Au−Auh‖V ∗‖u− vh‖V ≤ Γ(χ−1(‖f‖V ∗))‖u − uh‖V ‖u− vh‖V ,

from which we conclude the asserted estimate. Q.E.D.

Remark 1.6: The estimate (1.4.40) is another generalization of the “best-approximation” re-sult in the linear case where χ(t) = αt .

1.4.2 Application to the p-Laplace problem

Now, we use the abstract error estimate stated in Theorem 1.6 for the p-Laplace problem. For aquasi-uniform family of meshes Thh>0 of Ω let Vh ⊂ V be the usual finite element subspacesof piecewise linear elements.

First, we prove the existence of unique solutions for problem (1.4.36) and its discrete analogue(1.4.37). Recall the definition of the norm ‖v‖V := ‖∇v‖p of the Banach space V .

Theorem 1.7 (Existence of solutions): In case of the p-Laplacian operator, 2 ≤ p < ∞ ,problem (1.4.36) and its discrete analogue (1.4.37) possess unique solutions u ∈ V = H1,p

0 (Ω)and uh ∈ Vh , which are the unique minimizers of the corresponding functional J(·) on V andVh , respectively.


Proof: (i) For the functional J(·) there holds

J(v) = 1p‖v‖

pV − f(v) ≥ 1

p‖v‖pV − ‖f‖V ∗‖v‖V ,

from which we deduce that J(·) is bounded from below and that

J(v) → ∞ for ‖v‖ → ∞.

(ii) We show the strict convexity of J(·) . Since the functional f(·) is linear and thereforeconvex, it suffices to establish the strict convexity of the mapping

v ∈ V →∫

ΩF (∇v) dx, F : ξ ∈ R

d → 1p |ξ|p.

Let v and w be two different functions in V such that

meas Ω∗ > 0, Ω∗ := x ∈ Ω |∇v 6= ∇w

and let θ ∈ (0, 1) be given. Then, by the strict convexity of the mapping t ∈ R → |t|p,∫

ΩF (θ∇v + (1− θ)∇w) dx =

∫

Ω∗

... dx+

∫

Ω\Ω∗

... dx

<

∫

Ω

θF (∇v) + (1− θ)F (∇w)

dx.

We note at this stage that the strict convexity of the functional J(·) implies the uniqueness ofpossible minimizers on V and Vh .

(iii) We show the (Frechet) differentiability of .J(·) . The mapping F defined above is twicedifferentiable, with

∂iF (ξ) = |ξ|p−2ξi, ξ ∈ Rd,

∂i∂jF (ξ) = (p− 2)|ξ|p−4ξiξj , ξ ∈ Rd.

Consequently, we can write

F (ξ + η)− F (ξ) = |ξ|p−2ξ · η +R(ξ, η),

with bounded remainder term

|R(ξ, η)| ≤ c(p)(|ξ] + |η|)p−2|η|2.

Thus, ∫

ΩF (∇(u+ v)) dx−

∫

ΩF (∇u) dx =

∫

Ω|∇u|p−2∇u · ∇v dx+R(u, v),

with

|R(u, v)| ≤ c(p)

∫

Ω(|∇u|+ |∇v|)p−2|∇v|2 dx.

Since ∣∣∣∫

Ω|∇u|p−2∇u · ∇v dx

∣∣∣ ≤ ‖∇u‖p−1V ‖v‖V ,


the linear mapping

v ∈ V →∫

Ω|∇u|p−2∇u · ∇v dx

is continuous for a fixed u ∈ V . Further, since

∫

Ω(|∇u|+ |∇v|)p−2|∇v|2 dx ≤ (‖u||V + ‖v‖V )p−2‖v‖2V ,

this shows that the mapping considered in (ii) and by that also the functional J(·) is differen-tiable. Its derivative has the form

J ′(u)(v) =

∫

Ω|∇u|p−2∇u · ∇v dx− f(v), v ∈ V. (1.4.41)

Hence, the (unique) minimizers u and uh of J(·) on V and Vh must necessarily satisfy thevariational equations

J ′(u)(v) = 0 ∀v ∈ V, (1.4.42)

J ′(uh)(vh) = 0 ∀vh ∈ Vh. (1.4.43)

Further, in view of the convexity of the functional J(·) , any solutions of these equations arealso minimizers.

(iv) The “discrete” minimization problems have solutions. This is a consequence of the strictconvexity of the functional J(·) and the property J(vh) → ∞ for ‖vh‖V → ∞ . The argumentis analogous to that already used in the context of the minimal surface problem. Further, lettingvh = uh in the variational equation (1.4.43) gives us ‖uh‖pV = f(uh) and, consequently, theuniform bound

‖uh‖V ≤ ‖f‖1/(p−1)V ∗ .

(v) We use the previous results for extablishing the existence of a (unique) minimizer u ∈ V .The space V = H1,p(Ω), 1 < p < ∞, is reflexive. Then, the bounded set of approximateminimizers uh ∈ Vh contains a sequence (uhk)k ∈ N , with hk → 0 (k → ∞) , which convergesweakly to some element u ∈ V ,

ϕ ∈ V ∗ : ϕ(uhk) → ϕ(u), k → ∞.

We want to show that this limit u is a minimizer of J(·) . To this end let ψ ∈ C∞0 (Ω) be

arbitarily chosen and Ihkψ ∈ Vhk its nodal interpolant. For that there holds

J(uh) ≤ J(Ihkψ).

Since the functional J(·) is continuous and convex, it is weakly lower semicontinuous, andconsequently,

J(u) ≤ lim infk→∞J(uhk) ≤ lim infk→∞J(Ihkψ).

Observing that ‖ψ − Ihkψ‖V → 0 (k → ∞) by the continuity of J(·) it follows that

limk→∞

J(Ihkψ) = J(ψ),

and thusJ(u) ≤ J(ψ), ψ ∈ C∞

0 (Ω)


Since C∞0 (Ω) is dense in H1,p(Ω) it follows that u is indeed minimizer of J(·) . Q.E.D.

Corollary 1.3 (Qualitative convergence): The minimizers uh ∈ Vh converge to the mini-mizer u ∈ V = H1,p(Ω) :

‖uh − u‖V → 0 (h→ 0). (1.4.44)

Proof: In the proof of Theorem 1.7, we showed that there is a sequence (hhk)k∈N of discretesolutions, which converges weakly to the minimizer of J(·) , Since this limit is unique the wholefamily (uh)h>0 of discrete solutions converges weakly to this solution u ∈ V as h → 0 .Therefore,

f(u) = limh→0

f(uh).

Furthermore, we have for arbitrary ψ ∈ C∞0 (Ω) :

lim suph→0 J(uh) ≤ lim suph→0 J(Ihψ) = limh→0

J(Ihψ) = J(ψ).

Since C∞0 (Ω) is dense in V = H1,p(Ω) the function ψ can be chosen arbitrarily close to u in

the norm of V . Hence, we deduce that

J(u) ≤ lim infh→0J(uh) ≤ lim suph→0J(uh) ≤ J(u),

and consequently,J(u) = lim

h→0J(uh).

From this, we also get‖u‖V = lim

h→0‖uh‖V .

Since the space V = H1,p(Ω) is uniformly convex this implies the strong convergence

‖u− uh‖V → 0 (h→ 0).

Q.E.D.

Remark 1.7: In the previous proofs some deep results from abstract Functional Analysis havebeen used:

0) The Banach space H1,p(Ω), 1 < p <∞, is reflexive and uniformly convex.

1) In a reflexive Banach space bounded sets are weakly compact, i. e., contain weakly convergentsequences.

2) On a reflexive Banach space continuous and convex functionals are also weakly lower semi-continuous.

3) In a reflexive, uniformly convex Banach space a weakly convergent sequence (uk)kinN con-verges also strongly if the corrsponding sequence of norms converges, ‖uk‖ → ‖u‖ (k → ∞) .

For the proofs of these results the reader may consult the standard textbooks on FunctionalAnalysis, particularly those on Convex Analysis.

1.5 Exercises 33

Corollary 1.4 (Error estimate): For the finite element approximation of the p-Laplac prob-lem there holds the error estimate

‖u− uh‖1,p ≤ ch1/(p−1), (1.4.45)

with a constant c(‖f‖V ∗ , ‖u‖2,p) .

Proof: The operator A : V → V ∗ corresponding to the p-Laplacian operator is monotone withthe function χ(t) = αtp−1 . Hence the abstract error estimate of Theorem 1.6 yields

α‖u− uh‖p−1V ≤ c inf

vh∈Vh‖u− vh‖V .

Further,inf

vh∈Vh‖u− vh‖V ≤ ‖u− Ihu‖V ≤ ch‖u‖2,p,

which concludes the proof. Q.E.D.

Remark 1.8: The assumption u ∈ H2,p(Ω) made in Corollary 1.4 may not be realistic ingeneral situations. Solutions of nonlinear equations do not need to be smooth for smooth data(see exercise).

1.5 Exercises

Exercise 1.1: Consider the nonhomogeneous boundary value problem

−∆u = f in Ω, u = g on ∂Ω,

on a sufficiently regular domain Ω ⊂ Rn. The right-hand side f is in L2(Ω) and the boundary

function g is given as the trace of a function g ∈ H1(Ω) . The corresponding energy forms

a(u, v) := (∇u,∇v), E(u) := 12a(u, u) − (f, u),

are defined on the Sobolev space H1(Ω). Show by the arguments introduced in class that theminimization problem

minE(u) on Vg := v ∈ H1(Ω), v|∂Ω = g,

possesses a unique solution u , which as “weak solution” of the boundary value problem ischaracterized by the variational equation

a(u, ϕ) = (f, ϕ) ∀ϕ ∈ H10 (Ω).

Exercise 1.2: Consider the minimization problem

min(1p

∫

Ω

(1 + |∇u|2

)p/2dx−

∫

Ωfu dx

)on H1,p

0 (Ω),

for some p ∈ [1,∞) and a given function f ∈ L2(Ω). Derive the corresponding variationalformulation (first-order necessary optimality condition) and, in case of a sufficiently regularminimizer u , the resulting nonlinear boundary value problem.


Exercise 1.3: The Sobolev space H20 (Ω) is defined as the closure of the space C∞

0 (Ω) of allC∞-functions with compact support in Ω with respect to the norm

‖v‖H2 :=(‖∇2v‖2Ω + ‖∇v‖2Ω + ‖v‖2Ω

)1/2.

a) Show that this norm is equivalent to the semi-norm ‖∇2v‖Ω and on a convex (pogonal)domain also equivalent to the semi norm ‖∆v‖Ω .

b) Show that for functions in H20 (Ω) there holds

2(∂1∂2u, ∂1∂2v)Ω − (∂21u, ∂22v)Ω − (∂22u, ∂

21v)Ω = 0.

Exercise 1.4: Apply the existence theorem for convex minimization problems provided in classto show existence and uniqueness of solutions for the obstacle problem in plate bending theory,

minE(u) on Vψ := v ∈ H20 (Ω), v ≥ ψ

where, with some σ ∈ [0, 1] ,

E(u) := 12‖∆u‖2Ω + (1− σ)

(‖∂1∂2u‖2Ω − (∂21u, ∂

22u)Ω

)− (f, u),

and ψ is an admissible (i. e., positive only on a compact subset of Ω) smooth function describingan obstacle.

Exercise 1.5: Recall that the unique minimizer u ∈ M of the quadratic functional J(u) =12a(u, u) − l(u) on a convex, closed subset M of a Hilbert space V is characterized by theassociated variational inequality

a(u, v − u ≥ l(v − u) ∀v ∈M.

Show that if the subset M is a closed convex cone with vertex at the origin, i. e., u, v ∈ Mimplies αu + βv ∈ M for α, β ∈ R+, then the characterizing variational inequality takes thesimplified form

a(u, v) ≥ l(v) ∀v ∈M, a(u, u) = f(u).

An examples of such a cone is the set M = v ∈ H1(Ω) |u ≥ 0 a. e. on Ω ⊂ H1(Ω) .

Exercise 1.6: Consider the obstacle problem

min12‖∇u‖2 − (f, u)

on M := v ∈ H1

0 (Ω) | v ≥ ψ a. e. in Ω,

with given f ∈ L2(Ω) and obstacle ψ ∈ H10 (Ω)∩H2(Ω) . Show that the unique solution u ∈M

of this problem corresponds to the formal solution of the following boundary value problem:

−∆u ≥ f in Ω,

u ≥ ψ in Ω,

(−∆u− f)(u− ψ) = 0 in Ω,

u = 0 on ∂Ω,

u = ψ, ∂nu = ∂nψ on Γ,

1.5 Exercises 35

where Γ is the (unknown) “interface” (free boundary) between the subsets of Ω where theconstraint u ≥ ψ is active or inactive, respectively.

Exercise 1.7: Consider the special Signorini problem

minJ(u) on M := v ∈ V := H1(Ω) | v ≥ 0 on ∂Ω,

whereJ(v) := 1

2a(v, v) − (f, v), a(v, ϕ) = (∇v,∇ϕ) + (v, ϕ).

on a convex polygonal domain Ω ⊂ R2 . This problem is approximated by the finite element

method with piecewise linear elements on a quasi-uniform family of meshes Thh>0 :

minJ(uh) on Mh := vh ∈ Vh | vh ≥ 0 on ∂Ω.

Both problems, the continuous as well as the discrete ones, possess unique solutions, which arecharacterized by variational inequalities.

a) State the variational inequalities corresponding to this problem.

b) Derive a variant of the abstract V -error estimate developed in class, which is suited for thepresent special situation.

c) On the basis of this abstract error estimate prove the estimate

‖u− hh‖H1 ≤ c(f, u)h3/4.

Exercise 1.8: The error estimate ‖u− uh‖H1 ≤ c(u)h for the finite element approximation ofthe minimal surface problem has been proven in class for piecewise linear elements on quasi-uniform families of triangulations.

a) Convince yourself that the proof remains valid also for the case of mesh families not satisfyingthe uniform-size condition, which allows for local mesh refinement.

b) Discuss whether the proof carries over to the case of piecewise bilinear elements on quadri-lateral meshes.

Exercise 1.9: Show that the relation

Au(ϕ) :=

∫

Ω‖∇u‖p−2∇u · ∇ϕdx, ϕ ∈ V := H1,p(Ω),

for some p ∈ (1,∞), defines an operator A : V → V ∗ . For this use the general Holder inequalityfor (scalar) functions v ∈ Lp(Ω), w ∈ Lq(Ω) , 1/p + 1/q = 1 :

∣∣∫

Ωvw dx

∣∣∣ ≤( ∫

Ω|v|p dx

)1/p( ∫

Ω|w|q dx

)1/q, v ∈ Lp(Ω), w ∈ Lq(Ω).

Exercise 1.10: Consider the special one-dimensional p-Laplace problem on the domain Ω :=(−1, 1) where

J(v) :=1

p

∫

Ω|v′|p dx−

∫

Ωv dx.


Show that the unique minimizer u ∈ H1,p0 (Ω) of the functional J(·) is given by

u(x) = (1− 1/p)(1− |x|p/(p−1)

),

and that

u ∈ H2,p(Ω), 1 < p <3 +

√5

2, u 6∈ H2,p(Ω),

3 +√5

2≤ p <∞.

This demonstrates the possible limits in the regularity of solutions to strongly nonlinear prob-lems.

Exercise 1.11: Consider the approximation of the (linear) boundary value problem

−∆u = f in Ω, u = 0 on ∂Ω,

by “linear” finite elements. For this approximation we have the usual H1-norm error estimate

‖u− uh‖H1 ≤ ch‖u‖H2 ,

provided that u ∈ H10 (Ω) ∩H2(Ω) . Show in case that only the minimal regularity u ∈ H1

0 (Ω)is known that still qualitative convergence holds

‖u− uh‖H1 → 0 (h→ 0).

However, this convergence is not uniform with respect to ‖u‖H1 . (Hint: One may use the factthat the space C∞

0 (Ω) is by definition dense in H10 (Ω) .)

1.5 Exercises 37

2 General Quasilinear Elliptic Problems

In this chapter, we discuss the finite element approximation of general so-called “quasi-linear”elliptic boundary value problems in “divergence form”:

−d∑

i=1

∂iFi(·, u,∇u) + F0(·, u,∇u) = 0 in Ω, u = g on ∂Ω, (2.0.1)

in Rd , and to systems of such equations in which case u is a vector function. The material

of this chapter and further details can be found in the articles Frehse&Rannacher [39, 40],Dobrowolski &Rannacher [38], Rannacher [45, 47, 48, 52], and Rannacher&Scott [53].

The above setting includes the following special cases:

1. Minimal surface problem, with

Fi(·, u,∇u) := (1 + |∇u|2)−2/2∂i, F0(·, u,∇u) = 0,

2. Modified p-Laplace problem for 1 < p <∞ , with

Fi(·, u,∇u) := (γ + |∇u|2)(p−2)/2∂i, F0(·, u,∇u) = −f,

where γ > 0 , the limit case γ = 0 being excluded,

3. Nonlinear elasticity problem (geometrically and statically), with

Fi(·, u,∇u) :=d∑

j,k=1

(δjk + ∂kuj)sik, F0(·, u,∇u) = −f,

where the stress components sik are given in terms of the strain components εik =12(∂kui+∂iuk)+

12∂iuk∂kui via a strain energy functional Φ[ε(u)] (so-called “hyper-elastic”

material):

sik(u) =∂Φ

∂εik[ε(u)].

4. Nonlinear diffusion problem, with

Fi(·, u,∇u) := a(u)∂i, F0(·, u,∇u) = −f,

5. Nonlinear diffusion-transport problem (“vector Burgers equation”), with

Fi(·, u,∇u) := ν∂i, F0(·, u,∇u) = u · ∇u− f,

6. Nonlinear Diffusion-reaction problem, with

Fi(·, u,∇u) := D∂i, F0(·, u,∇u) = −f(u).

We emphasize that the considered problem does not need to originate from a minimizationproblem. In contrast to the treatment of the p-Laplace problem in Section 1.4, which was for-

39


mulated in an appropriate Sobolev (Banach) space H1,p(Ω) , here we prefer to use the standardSobolev (Hilbert) space V := H1

0 (Ω) but intersected with W 1,∞(Ω) . We will use again thenotation ‖ · ‖p := ‖ · ‖Lp and ‖ · ‖m,p := ‖ · ‖Hm,p for the norms of Lp(Ω), 1 ≤ p ≤ ∞, ,Hm,p(Ω), m ∈ N, 1 ≤ p < ∞, , and Wm,∞(Ω) , respectively. The scalar product and norm ofL2(Ω) are written without subscript as (·, ·) and ‖ · ‖ . This notation is also used with theobvious interpretation for vector and tensor functions, e. g., (∇v,∇w), ‖∇v‖1 , ‖∇2v‖∞, etc. .

2.1 Quasi-linear problems

For technical simplicity, we only consider the special situation of a (convex) polygonal domainΩ ⊂ R

2 and homogeneous Dirichlet data g = 0 . The case of nonhomogeneous Dirichlet data,which would be required in treating the minimal surface problem can be covered by the standardtechnical modifications and all results presented below remain balid for this case. Since theanalysis for problem (2.0.1) in its full generality is rather technical and lengthy, for clarity ofpresentation, we prefer to restrict the following analysis to the prototypical special case of scalarproblems in R

2 of the form

−∇ · F (·,∇u) = f in Ω, u = 0 on ∂Ω. (2.1.2)

For the treatment of more general situations, we refer to the literature [40], [38], and [47, 48],from which much of the contents of this chapter is taken.

The function F (x, η) is assumed to be sufficiently regular (i. e. differentiable) with respectto all its arguments x ∈ Ω and η ∈ R

2 . In particular, its Jacobian matrix F ′(·, η) = ∇ηF (·, η)is positive definite and Lipschitz continuous for bounded arguments, uniformly for x ∈ Ω ,

(F ′(·, η)ξ, ξ) ≥ α(η)|ξ|2, ξ ∈ R2, |F ′(·, η) − F ′(·, η′)| ≤ γ(η, η′)|η − η′|, η, η′ ∈ R

2,

with a constant α(η) > 0 .

The variational formulation of problem (2.1.2) reads as follows:

(P) Find a function u ∈ V ∩W 1,∞(Ω) , such that

a(u;ϕ) = (f, ϕ) ∀ϕ ∈ V, (2.1.3)

where we use the semi-linear form (nonlinear in its first and linear in its second argument)

a(u;ϕ) := (F (·,∇u),∇ϕ),

which is well-defined on (V ∩W 1,∞(Ω))× V .

We are not interested here in the question of existence of solutions of problem (2.1.3) butin their numerical approximation by the finite element Galerkin method. Therefore, we simplypresuppose the existance of a solution u ∈ V ∩W 1,∞(Ω) , which is unique in a certain W 1,∞-ball

BR(u) = v ∈ V ∩W 1,∞(Ω), |||u− v|||∞ ≤ R, |||v|||∞ := ess supΩ|∇v|.

The local uniqueness of the solution u is related to the assumption that the “tangent form”

a′(u; v,w) := (F ′(·,∇u)∇v,∇w), v, w ∈ V,

2.1 Quasi-linear problems 41

corresponding to the semilinear form a(·; ·) is uniformly V -elliptic for arguments v ∈ BR(u) :

a′(v;ϕ,ϕ) ≥ α(R)‖∇ϕ‖2, ϕ ∈ V. (2.1.4)

Further, by our assumptions on F (·, ·) the tangent form a′(·; ·, ·) has the following Lipschitzcontinuity property, uniformly on BR(u) ,

|a′(u;ϕ,ψ) − a′(v;ϕ,ψ)| ≤ c(R)|||u− v|||∞|||ϕ||∞|‖∇ψ‖1, (2.1.5)

for v ∈ V ∩ BR(u), ϕ ∈ V ∩ W 1,∞(Ω), ψ ∈ V . The data of the problem, particularly thedomain Ω , should be sufficiently regular, such that for some q > 2 the linear operator A′(u) :V ∩H2,q(Ω) → Lq(Ω) generated by the bilinear form a′(u; ·, ·) on V is onto and satisfies thea priori estimate

‖v‖2,q ≤ c‖A′(u)v‖q. (2.1.6)

The link between the bilinear form a′(u; ·, ·) and the operator A′(u) is given by the relation

(A′(u)v, ϕ) := a′(u; v, ϕ), v, ϕ ∈ V.

It is known that this condition is satisfied particularly on convex polygonal domains in R2 .

Remark 2.1: We have avoided making any assumption which would guarantee the global mono-tonicity of problem (2.1.3). Only local regularity and continuity properties are required in aW 1,∞-neighborhood of a sufficiently regular solution. Clearly our assumptions exclude prob-lems with solutions having unbounded gradients, e. g., in the neighborhood of reentrant cornersor at points where the ellipticity degenerates.

2.1.1 Finite element discretization

For the discretization of problem (2.1.3), we consider only the simplest finite element Galerkinmethod. Let again Thh>0 be a quasi-uniform family of triangulations covering the polygonaldomain Ω and Vh ⊂ V the usual subspaces of piecewise linear functions:

Vh = vh ∈ V, vh|T ∈ P1(T ), T ∈ Th ⊂W 1,∞(Ω).

Then, the discrete problems read as follows:

(Ph) Find uh ∈ Vh , such that

a(uh;ϕh) = (f, ϕh) ∀ϕh ∈ Vh. (2.1.7)

The existence of solutions to the discrete problems (Ph) will be obtained from the assumedexistence of a solution to the continuous problem (P).

Remark 2.2: For the following, we have assumed the mesh family Thh>0 to be “quasi-uniform”. i. e., particularly to satisfy the “uniform size” and “uniform shape” condition. Thisassumption is mainly for technical convenience. Most of the presented results remain essentiallyvalid if only a “weak size-uniformity” condition (for the precise definition see below) is satisfied,which allows for local mesh refinement. This is important if systematic mesh adaptivity is used


in solving such nonlinear problems. We will address the question of required mesh regularityexplicitly below.

Theorem 2.1: Let the above conditions be satisfied, particularly let problem (P) possess a so-lution u ∈ V ∩ H2,p(Ω) for some p > 2 , which is unique in some W 1,∞-ball BR(u). Then,for sufficiently small h, h ≤ h0 , the discrete problems (Ph) also possess locally unique solutionsuh ∈ Vh ∩BR(u) and there holds the basic W 1,∞-error estimate

|||u− uh|||∞ ≤ c(u)L(h)h1−2/p, 2 < p ≤ ∞, (2.1.8)

where L(h) := maxlog(1/h), 1 . Further, under the same assumptions there hold the improvedL∞- and L2-error estimates

‖u− uh‖∞ ≤ c(u)h2−2/pL(h), 2 < p ≤ ∞, (2.1.9)

‖u− uh‖2 ≤ c(u)h2. (2.1.10)

Proof: The proof employs a so-called “homotopy argument”. In this argument the givennonlinear problems are embedded into a parameter-dependent family of problems containing,for one special parameter value, a linear problem, for which the asserted error estimate is known.By a continuation argument that estimate is then carried over to the given nonlinear problems.We shall give the details only for the basic W 1,∞-error estimate (2.1.8) leave the modificationof the argument for covering also the other estimates as an exercise. Below, the symbol c willbe used as a “generic” constant, which may vary with the context, but is always independent ofall critical parameters involved.

(i) We begin with some technical preliminaries. We introduce the linear “Ritz projection”Rh : V → Vh defined by

a′(u;Rhv, ϕh) = a′(u; v, ϕh) ∀ϕh ∈ Vh, v ∈ V. (2.1.11)

This construction is well-defined, since by assumption the bilinear form a′(u; ·, ·) is continuousand V -elliptic (Lax-Milgram theorem). Under our assumptions, this Ritz projection is knownto be almost W 1,∞-stable, i. e., there holds the estimate

|||Rhv|||∞ ≤ cL(h)|||v|||∞ v ∈ V ∩W 1,∞(Ω). (2.1.12)

This stability estimate will be proven in the next section. In view of the usual approximationestimate for the nodal interpolation Ih : V ∩C(Ω) → Vh ,

|||u− Ihu|||∞ ≤ ch1−2/p‖u‖2,p, 2 < p ≤ ∞, (2.1.13)

the stability estimate (2.1.12) for Rh implies the error estimate

|||u−Rhu|||∞ ≤ |||u− Ihu|||∞ + |||Rh(Ihu− u)|||∞≤ cL(h)|||u − Ihu|||∞ ≤ c0L(h)h

1−2/p‖u‖2,p, 2 < p ≤ ∞.(2.1.14)

Further, the following error estimates are known for the Ritz projection:

‖u−Rhu‖∞ ≤ ch2−2/pL(h)‖u‖2,p, 2 < p ≤ ∞, (2.1.15)

‖u−Rhu‖2 ≤ ch2‖u‖2,2. (2.1.16)


These can be deduced similarly as in (2.1.14) from the stability estimates

‖Rhv‖∞ ≤ c‖v‖∞ + chL(h)‖∇v‖∞, v ∈ V ∩W 1,∞(Ω), (2.1.17)

‖Rhv‖ ≤ c‖v‖+ ch‖∇v‖, v ∈ V ∩H2(Ω), (2.1.18)

which are among the estimates proven in the next section. The Ritz projection Rh : V → Vhcan be extended to an operator (using the same notation) Rh : V ∗

h → Vh by defining

a′(u; Rhv∗h, ϕh) = v∗h(ϕh) ∀ϕh ∈ Vh, v∗h ∈ V ∗

h .

Notice that in the usual sense V ∗h ⊂ V ∗ = V . For this extended Ritz projection there holds the

stability estimate

|||Rhv∗h|||∞ ≤ cL(h)‖v∗h‖∞;h, (2.1.19)

where the dual norm ‖ · ‖∞;h is defined by

‖v∗h‖∞;h := supv∗h(vh) | vh ∈ Vh, ‖∇vh‖1 = 1

.

This stability estimate will also be proven in the next section. The logarithmic factor L(h)in the stability estimate (2.1.19) cannot be avoided, which can be shown by counter examples(exercise). In this analysis, we do not make use of the uniform size property of the mesh familyThh>0 , only the uniform shape property is required.. Therefore all results in this chapter holdon such mesh families, which particularly allows for local mesh refinement. It can be shownthat for quasi-uniform meshes the extra logarithmic factor L(h) in the W 1,∞-stability estimate(2.1.12) and also in the W 1,∞-error estimates (2.1.14) and consequently in the correspondingestimate (2.1.8) in the main Theorem 2.1 does not occur (see [53]).

(ii) In order to carry the estimate (2.1.14) over to the nonlinear problem (2.1.3), we use ahomotopy argument. For a homotopy parameter t ∈ [0, 1] , we introduce the semilinear forms

at(v;w) := ta(v;w) + (1− t)a′(u; v − u,w)

and the corresponding auxiliary problems

(P t) Find ut ∈ V ∩W 1,∞(Ω) such that

at(ut; v) = t(f, v) ∀v ∈ V, (2.1.20)

and their discrete analogues

(P th) Find uth ∈ Vh such that

at(uth; vh) = t(f, vh) ∀vh ∈ Vh. (2.1.21)

Clearly, u1h = uh and u0h = Rhu . Further, for all t ∈ [0, 1] , the function ut := u is a solutionof problem (2.1.20) and there holds a′0(u; ·, ·) = a′(u; ·, ·) . We define the set

Θh :=t ∈ [0, 1] |Problem (P th) has a locally unique solution uth ∈ Vh ∩BR(u),for which there holds |||u− uth|||∞ < 2c0L(h)h

1−2/p.,


where c0 is the constant in the “linear” error estimate (2.1.14). We want to show that, forsufficiently small h, h ≤ h0 , the set Θh is nonempty, open and closed with respect to [0, 1]and therefore coincides with [0, 1] . Then, 1 ∈ Θh implies the asserted error estimate.

(iii) By definition, we have 0 ∈ Θh so that Θh 6= ∅ .(iv) Next, we establish the closedness of Θh for any fixed h , which is the harder part ofthe proof. Consider a convergent sequence (tk)k∈N ⊂ Θh , with t := limk→∞ tk , and thecorresponding sequence (utkh )k∈N of discrete solutions utkh ∈ Vh ∩ BR(u) . Since the sequence

(utkh )k∈N is bounded in the finite dimensional space Vh , there exists a convergent subsequencewith limit uth ∈ Vh ∩ BR(u) . By continuity uth is a solution of the corresponding variationalproblem (P th), for which the error estimate holds

|||u− uth|||∞ ≤ 2c0L(h)h1−2/p.

On Vh the tangent form of at(·; ·) is given by

a′t(vh;wh, ϕh) := ta′(vh;wh, ϕh) + (1− t)a′(u;wh, ϕh), vh, wh, ϕh ∈ Vh,

which clearly is V -elliptic for vh ∈ BR(u) . This implies that the discrete solution uth is locallyunique. Since for any other solution uth ∈ Vh ∩BR(u) the relation

0 = at(uth; vh)− at(u

th; vh) =

∫ 1

0a′t(u

th + t(uth − uth);u

th − uth, vh) dt,

observing that uth + t(uth − uth) ∈ BR(u) and setting vh = uth − uth implies uth = uth . Further,observing that uth solves

at(uth; vh) = ta(uth; vh) + (1− t)a′(u;uth − u, vh) = t(f, vh) ∀vh ∈ Vh,

we can write

a′(u;u− uth, vh) + ta(u; vh)− ta(uth; vh) = a′(u;u− uth, vh) + t(f ; vh)− ta(uth; vh)

= a′(u;u− uth, vh) + (1− t)a′(u;uth − u, vh)

= ta′(u;u − uth, vh)

and therefore,

a′(u;Rhu− uth, vh) = a′(u;u− uth, vh) + ta(u; vh)− a(uth; vh)

− t

a(u; vh)− a(uth; vh)

= ta′(u;u− uth, vh)− t

∫ 1

0a′(uth + s(u− uth);u− uth, vh) ds

= ta′(u;u− uth, vh)− t

∫ 1

0a′(uth + s(u− uth);u− uth, vh) ds

= t

∫ 1

0

a′(u;u− uth, vh)− a′(uth + s(u− uth);u− uth, vh)

ds.

The right-hand side of this equation can be viewed as a functional v∗h ∈ V ∗h acting on vh ∈ Vh ,

i. e., Rhu − uth = Rhv∗h . By the assumed Lipschitz continuity of a′(·;u − uth, vh) for bounded


arguments, the dual norm of this functional is estimate by

‖v∗h‖∞;h ≤ ct|||u− uth|||2∞.

Consequently, by the stability estimate (2.1.19),

|||Rhu− uth|||∞ ≤ cL(h)‖v∗h‖∞;h ≤ ctL(h)|||u − uth|||2∞.

Hence, by the error estimate (2.1.14), we obtain

|||u− uth|||∞ ≤ |||u−Rhu|||∞ + |||Rhu− uth|||∞≤ c0L(h)h

1−2/p + c1tL(h)|||u − uth|||2∞,(2.1.22)

with constants c0 and c1 independent of h and θ . This implies that for h ≤ h0 sufficientlysmall

|||u− uth|||∞ < 2c0L(h)h1−2/p,

i. e., t ∈ Θh .

(iv) For proving the openness of Θh , we employ the implicit function theorem. For any t ∈ Θh

we define a function H(t, ϕ) : [0, 1] × (Vh ∩BR(u)) → V ∗h (dual space of Vh ) by setting

H(t, ϕ)(·) := ta(ϕ; ·) + (1− t)a′(u;ϕ− u, ·)− t(f, ·).

Obviously, H(t, ϕ) is continuous and continuously differentiable with respect to ϕ on [0, 1] ×(Vh ∩BR(u)) with derivative

H ′(t, ϕ)(ψ, ·) = ta′(ϕ;ψ, ·) + (1− t)a′(u;ψ, ·), ψ ∈ Vh.

Then, for any τ ∈ Θh , we have by definition,

uτh ∈ Vh ∩BR(u), H(τ, uτh)(·) = 0.

andH ′(τ, uτh)(ψ, ·) = τa′(uτh;ψ, ·) + (1− τ)a′u;ψ, ·), ψ ∈ Vh.

By the V -ellipticity the relation

H ′(τ, uτh)(ψ,χ) = 0, χ ∈ Vh,

implies that necessarily ψ = 0 . Hence the inverse H ′(τ, ψ)−1 exists since Vh is finite di-mensional. Then, the implicit function theorem, applied to H(t, ϕ) , yields the existence of aneighborhood N(τ) ⊂ [0, 1] such that there are functions ϕ(t) ∈ Vh ∩BR(u) , which satisfy

H(t, ϕ(t))(·) = 0, t ∈ N(τ),

and are continuous with respect to t ∈ N(τ) . This obviously shows that t ∈ Θh for |t − τ |sufficiently small. Hence, Θh is open and the proof is complete. Q.E.D.


2.1.2 Auxiliary L∞-stability estimates for the linearized problems

In the following, we give the proofs of the stability estimates (2.1.12) and (2.1.19) for the Ritzprojections Rh and Rh introduced above in the proof of Theorem 2.1. Since this has expositorycharacter, we restrict the presentation to the model problem of the Poisson equation:

−∆u = f in Ω, u = 0 on ∂Ω, (2.1.23)

on a convex polygonal domain Ω ⊂ R2 . The argument for the more general elliptic operator

A′(u) is similar but technically more involved. Details can be found in the literature [39], [53],and [?]. Further, we only consider low-order piecewise linear finite elements on a mesh famililyThh>0 assumed to be strongly shape-regular but not in all cases necessarily strongly size-regular. We shall try to avoid as much as possible the use of the “uniform-size property” inorder to allow for local mesh refinement. Accordingly, we set h := maxT∈Th

hT (hT the radiusof the minimal circumscribed circle of T ) and ρ := minT∈Th

ρT (ρT radius of the maximalinscribed circle of T ).

Definition 2.1: The family of triangulations Thh>0 is said to be “weakly size-regular”, ifthere exist some α ≥ 1, such that uniformly for all meshes:

hmin ≥ chαmax, Th ∈ Thh>0. (2.1.24)

Remark 2.3: In practice the case 1 ≤ α ≤ 2 is most relevant in mesh adaptation. Faster meshrefinement is rarely necessary in standard situations. The main feature of “weak size regularity”is that in this case the logarithmic factor log(1/hmin) frequently occurring in error estimatesbehaves like α log(1/hmax) . In the extreme case of refinement towards a point by m-times localbisection, leading to hmin ∼ 2−mhmax , there would hold log(2m/hmax) ∼ m log(2)+log(1/hmax) ,meaning a significant loss in accuracy.

First, we consider the standard Ritz projection Rh : V → Vh , which in the present situationis defined by

(∇Rhu,∇ϕh) = (∇u,∇ϕh) ∀ϕh ∈ Vh. (2.1.25)

We note the following L2-stability estimate

‖Rhu‖ ≤ ‖u‖+ ch‖∇u‖, u ∈ V, (2.1.26)

which is easily obtained using a standard duality argument (exercise). The following lemmaprovides the corresponding L∞-stability estimate.

Theorem 2.2 (L∞ stability): Suppose that the domain Ω ⊂ R2 is convex polygonal and that

the regular mesh family Thh>0 is strongly shape- but possibly only weakly size-regular with anexponents α ≥ 1. Then, there holds

‖Rhu‖∞ ≤ c‖u‖∞ + cαhL(h)‖∇u‖∞, (2.1.27)

for functions u ∈ H10 (Ω) ∩W 1,∞(Ω) , with a constant c independent of u, h, and α.


Proof: We set uh := Rhu. Let T∗ be an arbitrary cell of a mesh Th with diameter h∗ = hT∗and maximal inscribed circle B∗ ⊂ T∗ with center point x∗ and radius ρ∗ = ρT∗ . By a standardscaling argument and the equivalence of norms on the finite dimensional space P1 there holds

‖uh‖∞;B∗≤ c|B∗|−1

∫

B∗

|uh| dx. (2.1.28)

The integral on the right of (2.1.28) can be rewritten as

|B∗|−1

∫

B∗

|uh| dx = (uh, δ),

where the regularized Dirac function δ := δh ∈ L2(Ω) is defined by

δ := |B∗|−1sign(uh) in B∗, δ := 0 else.

To the function δ , we associate a regularized Green’s function g = gh ∈ V as the solution ofthe dual problem

(∇ϕ,∇g) = (ϕ, δ) ∀ϕ∈V, (2.1.29)

and its Ritz projection gh := Rhg ∈ Vh . Then, by definition of uh and by Galerkin orthogo-nality, there holds

(uh, δ) = (∇uh,∇g) = (∇uh,∇gh) = (∇u,∇gh)= (∇u,∇(gh − g)) + (∇u,∇g) = (∇u,∇(gh − g)) + (u, δ),

and consequently,

‖uh‖∞;B∗≤ ‖∇u‖∞‖∇(g − gh)‖1 + ‖u‖∞;B∗

. (2.1.30)

Hence the proof is completed by proving the following Lemma 2.1 and observing that (exercise),

‖uh‖∞;T∗ ≤ c‖uh‖∞;B∗,

and, by the assumed strong shape- and weak size-regularity,

L(ρ∗) ≤ L(ch∗) ≤ L(chαmax) ≤ cαL(hmax) = cαL(h).

Q.E.D.

Lemma 2.1: For the regularized Green’s function there holds the H1,1-error estimate

‖∇(g − gh)‖1 ≤ chL(ρ∗). (2.1.31)

The proof of this lemma requires some preparation. We define the weight function

σ(x) := (|x− x∗|2 + κ2ρ2∗)1/2,


with some parameter κ ≥ 1, which will be appropropriately fixed below. By elementary calcu-lation, we see that

σ ≥ κρ∗, |∇σ| ≤ 1, ‖σ−1‖ ≤ cL(ρ∗)1/2, (2.1.32)

and for each mesh cell T ∈ Th,

maxT

σ ≤ minTσ + hT max

T|∇σ| ≤ min

Tσ + hT . (2.1.33)

The assertion of the following auxiliary lemma is crucial for the proof of Lemma 2.1.

Lemma 2.2: For fixed κ ≥ 1 there hold the following a priori estimates:

‖g‖∞ ≤ cL(ρ∗), (2.1.34)

‖∇g‖ + ‖σ∇2g‖ ≤ cL(ρ∗)1/2, (2.1.35)

‖∇2g‖ ≤ cρ−1∗ . (2.1.36)

Proof: (i) The true Green’s function Gx on Ω corresponding to an arbitrary point x ∈ Ωadmits the well-known estimate

|Gx(y)| ≤ c| ln(|y − x|)|+ 1,

which may be derived by using the maximum principle. Then, from the estimate (exercise)

|g(x)| = |(∇g,∇Gx)| = |(δ,Gx)| ≤ |B∗|−1

∫

B∗

|Gx| dy ≤ cL(ρ∗),

we obtain (2.1.34).(ii) Observing that

‖∇g‖2 = (δ, g) ≤ ‖g‖∞ ≤ cL(ρ∗),

we obtain the first part of (2.1.35). Further, by the usual H2 a priori estimate, we obtain(2.1.36),

‖∇2g‖ ≤ c‖∆g‖ = c‖δ‖ ≤ cρ−1∗ .

(iii) Next, we set ξ := x− x∗ and find

|ξi∇2g| ≤ |∇2(ξig)| + |∇g|,

and consequently,

‖σ∇2g‖2 =

2∑

i=1

‖ξi∇2g‖2 + κ2ρ2∗‖∇2g‖2

≤ c

2∑

i=1

‖∇2(ξig)‖2 + ‖∇g‖2

+ κ2ρ2∗‖∇2g‖2.


By the usual H2 a priori estimate,

‖∇2(ξig)‖ ≤ ‖∆(ξig)‖ ≤ ‖ξi∆g‖+ ‖∇g‖= ‖ξiδ‖ + ‖∇g‖ ≤ c+ cL(ρ∗)

1/2.

Combining the foregoing estimates, we obtain the second part of (2.1.35),

‖σ∇2g‖2 ≤ cL(ρ∗)1/2,


Proof of Lemma 2.1: Let η := g − gh . From the L2-stability estimate (2.1.26), which alsoholds on the general meshes considered, or alternatively by directly using the V -stability of theRitz projection and applying a duality argument, we obtain the well-known L2-error estimate

‖v −Rhv‖+ h‖∇(v −Rhv)‖ ≤ ch‖∇v‖, v ∈ V. (2.1.37)

Further, we recall the usual cellwise estimate for the standard nodal interpolation:

‖v − Ihv‖T + hT ‖∇(v − Ihv)‖T ≤ ch2T ‖∇2v‖T , T ∈ Th, v ∈ H2(T ). (2.1.38)

(i) We fix κ = 1. Combining the L2-error estimate (2.1.37) for v := g with the a priori estimate(2.1.35), we have

‖η‖ + h‖∇η‖ ≤ ch‖∇g‖ ≤ chL(ρ∗)1/2. (2.1.39)

Further there holds

‖∇η‖1 ≤ ‖σ−1‖ ‖σ∇η‖ ≤ cL(ρ∗)1/2‖σ∇η‖. (2.1.40)

For the term on the right, we have

‖σ∇η‖2 = (∇η,∇(σ2η)− (∇η, η∇σ2) =: E1 − E2.

The terms E1 and E2 will be estimated separately. First, using Galerkin orthogonality, we get

E1 = (∇η,∇(σ2η − ψh))

with the nodal interpolation ψh := Ih(σ2η) ∈ Vh . This term is estimated further using the

cellwise interpolation estimate (2.1.38),

E1 ≤∑

T∈Th

‖∇η‖T ‖∇(σ2η − ψh)‖T ≤ c∑

T∈Th

hT ‖∇η‖T ‖∇2(σ2η)‖T .

For the second factors on the right, we have

‖∇2(σ2η)‖T ≤ c‖η‖T + ‖σ∇η)‖T + ‖σ2∇2g‖T

,

and consequently,

E1 ≤ c∑

T∈Th

hT

‖∇η‖T

‖η‖T + ‖σ∇η‖T

+ ‖∇η‖T ‖σ2∇2g‖T

.


In view of the relation maxT σ ≤ minT σ + hT , we have

‖∇η‖T ‖σ2∇2g‖T ≤ maxTσ ‖∇η‖T ‖σ∇2g‖T≤

‖σ∇η‖T + hT ‖∇η‖T

‖σ∇2g‖T .

Hence, by Schwarz’s inequality,

E1 ≤ ch‖∇η‖‖η‖ + ‖σ∇η‖

+ ch

‖σ∇η‖+ h‖∇η‖

‖σ∇2g‖.

In view of the L2-error estimate (2.1.39) and the a priori estimate (2.1.35), we conclude

E1 ≤ 14‖σ∇η‖2 + ch2L(ρ∗).

For the second term E2 , we analogously obtain

E2 ≤ c‖σ∇η‖‖η‖ ≤ 14‖σ∇η‖2 + ch2L(ρ∗).

Finally, combining the estimates for E1 and E2 , we obtain

‖σ∇η‖2 ≤ 12‖σ∇η‖2 + ch2L(ρ∗),

which in view of (2.1.40) completes the proof of Lemma 2.1. Q:E:D.

Remark 2.4: We note that in the foregoing argument, we did not use the “strong shape regu-larity” of the mesh family Thh∈R+

, i. e., in 2D the L1 error estimate (2.1.31) holds true alsounder the assumption of only “weak shape regularity”, which is defined similarly as “weak sizeregularity”.

Remark 2.5: The logarithmic factor in the stability estimate (2.1.27) is unavoidable in general.This can be demonstrated by analytical arguments for special situations and is also confirmedby numerical experiments.

Next, we derive W 1,∞-stability estimates. We note that in case that the mesh family isquasi-uniform, a W 1,∞-stability estimate could be deduced (exercise) from the estimate (2.1.27)of Theorem 2.2 by using the inverse relation

‖∇vh‖∞ ≤ ch−1min‖vh‖∞, vh ∈ Vh,

together with the interpolation estimate

‖v − Ihv‖∞ + hmax‖∇(v − Ihv)‖∞ ≤ chmax‖∇v‖∞, v ∈W 1,∞(Ω).

However, this argument cannot be used for the extended Ritz projection Rh , for which the proofof W 1,∞-stability requires more work, even on quasi-uniform mesh families. In the followingargument, we will make this regularity assumption for technical simplicity. The extension toonly weakly size-uniform meshes, which would be very desirable in the context of adaptivemethods, has not been fully accomplished yet. We will comment on this aspect below in theproof of the following theorem.


Theorem 2.3 (W 1,∞-stability): Suppose that the domain Ω ⊂ R2 is convex polygonal and

that the regular mesh family Thh>0 is quasi-uniform. Then, there hold the W 1,∞-stabilityestimates

‖∇Rhu‖∞ ≤ cL(h)‖∇u‖∞, (2.1.41)

for u ∈ H10 (Ω) ∩W 1,∞(Ω) , and

‖∇Rhv∗h‖∞ ≤ cL(h)‖v∗h‖∞;h, (2.1.42)

for v∗h ∈ V ∗h , with constants c independent of u and h.

Proof: We continue using the notation of the proof of Theorem 2.2 and particularly that ofLemma 2.1. We set again uh := Rhu or uh := Rhv

∗h .

(i) For any fixed h > 0 let T∗ ∈ Th be an arbitrary cell with maximal inscribed circle B∗ :=Bρ∗(x∗) , with radius ρ∗ and center point x∗ , and h∗ := hT∗ . We will again use the weight

function σ(x) :=(|x− x∗|2 + κ2ρ2∗

)1/2, which satisfies

σ ≥ κρ∗, |∇σ| ≤ 1, ‖σ−1‖ ≤ cL(ρ∗)1/2. (2.1.43)

Now, there exists a Dirac-like function δ ∈ C∞0 (B∗) with the properties

0 ≤ δ ≤ cρ−2∗ , |∇δ| ≤ cρ−3

∗ ,

∫

B∗

δ dx = 1.

Since ∇uh is constant on T∗ , we have

‖∂iuh‖∞;T∗ = |(∂iuh, δ)|, i = 1, 2.

To the function δ , we associate regularized (“derivative”) Green’s functions g′i ∈ V, i = 1, 2 ,as the solution of the “dual problems”

(∇ϕ,∇g′i) = (∂iϕ, δ) ∀ϕ ∈ V, (2.1.44)

and their Ritz projections Rhg′i ∈ Vh . Then, by definition of uh and Galerkin orthogonality,

there holds in the case uh = Rhu ,

(∂iuh, δ) = (∇uh,∇g′i) = (∇uh,∇Rhg′i) = (∇u,∇Rhg′i)= (∇u,∇(Rhg

′i − g′i)) + (∇u,∇g′i) = (∇u,∇(Rhg

′i − g′i)) + (∂iu, δ),

and, consequently,

‖∂iuh‖∞;T∗ ≤ ‖∇u‖∞ maxi=1,2

‖∇(Rhg′i − g′i)‖1 + ‖∇u‖∞;T∗ . (2.1.45)

In the case uh = Rhv∗h , there holds

(∂iRhv∗h, δ) = (∇Rhv∗h,∇g′i) = (∇Rhv∗h,∇Rhg′i) = v∗h(Rhg

′i),

and, consequently,‖∂iRhv∗h‖B∗;∞ ≤ ‖v∗h‖∞;h‖∇Rhg′i‖1.


Hence using the estimate in the following Lemma 2.3 and observing again that, by the assumedproperties of the considered meshes,

L(ρ∗) ≤ L(ch∗) ≤ L(chmax) ≤ cL(h), (2.1.46)

the proof is completed. Q.E.D.

In the following, we drop the index i ∈ 1, 2 in the derivative Green’s functions and setg′ := g′i and g′h := Rhg

′ .

Lemma 2.3: For the regularized derivative Green’s functions g′ there holds the H1,1 estimate

‖∇(g′ − g′h)‖1 + ‖∇g′h‖1 ≤ cL(ρ∗). (2.1.47)

Proof: For abbreviation we set η′ := g′ − g′h . The proof is given in a sequence of steps.

(i) There holds

‖∇η′‖1 ≤ ‖σ−1‖‖σ∇η′‖ ≤ cL(ρ∗)1/2‖σ∇η′‖. (2.1.48)

For the term on the right, we have

‖σ∇η′‖2 = (∇η′,∇(σ2η′))− (∇η′, η′∇σ2) =: E′1 − E′

2.

The terms E′1 and E′

2 will be estimated separately. First, using Galerkin orthogonality, we get

E′1 = (∇η′,∇(σ2η′ − ψh))

with the nodal interpolant ψh := Ih(σ2η′) ∈ Vh . This term is estimated further, using the usual

cellwise L2-interpolation estimate and the quasi-uniformity of the meshes,

E′1 ≤

∑

T∈Th

‖∇η′‖T ‖∇(σ2η′ − ψh)‖T

≤ c∑

T∈Th

hT ‖∇η′‖T ‖∇2(σ2η′)‖T

≤ cκ−1∑

T∈Th

‖σ∇η′‖T‖η′‖T + ‖σ∇η′‖T + ‖σ2∇2g′‖T

≤ cκ−1‖σ∇η′‖‖η′‖+ ‖σ∇η′‖+ ‖σ2∇2g′‖

.

Remark 2.6: We note that the quasi-uniformity of the mesh family Thh>0 is used for theestimate

hT ‖∇η′‖T ≤ cκ−1‖σ∇η′‖T , (2.1.49)

which requires that κhT ≤ cminT σ, T ∈ Th . We conjecture that this is remains true for onlyweakly size-regular meshes, which would then imply the assertion of the theorem also under thiaweaker assumption.


For the next step, we need to estimate ‖η′‖ by ‖σ∇η′‖ . For this, we use a duality argument.Let z ∈ V be the solution of the auxiliary problem

(∇ϕ,∇z) = (ϕ, η′)‖η′‖−1 ∀ϕ ∈ V,

satisfying z ∈ H2(Ω) and ‖z‖2,2 ≤ c . Then, there holds

‖η′‖ = (∇η′,∇z) = (∇η′,∇(z − Ihz)),

and further‖η′‖ ≤ c

∑

T∈Th

hT ‖∇η′‖T ‖∇2z‖T ≤ cκ−1‖σ∇η′‖.

Now, using the just proven estimate ‖η′‖ ≤ cκ−1‖σ∇η′‖ , we conclude that

E1 ≤ cκ−1‖σ∇η′‖2 + c‖σ2∇2g′‖2.

For the second term E2 , we have

E2 ≤ c‖σ∇η′‖‖η′‖ ≤ cκ−1‖σ∇η′‖2.

Combining this with the above estimate for E′1 , we obtain

‖σ∇η′‖2 ≤ cκ−1‖σ∇η′‖2 + c‖σ2∇2g′‖.

Hence, in view of the estimate (2.4) in Lemma 2.4, below, chosing κ sufficiently large, we provethe main assertion of the lemma. Then, the complete assertion follows by using the estimate

‖∇g′h‖1 ≤ ‖∇(g′h − g′)‖1 + ‖∇g′‖1 ≤ cL(h)1/2‖σ∇(g′h − g′)‖+ ‖σ∇g′‖

,

again together with the a priori bound (2.1.50). Q.E.D.

Lemma 2.4: There holds the a priori estimate

‖σ∇g′‖+ ‖σ2∇2g′‖ ≤ cL(ρ∗)1/2. (2.1.50)

Proof: Using the usual H2-a priori estimate on Ω , we estimate as follows:

‖σ2∇2g′‖ ≤ ‖∇2(σ2g′)‖+ c‖σ∇g′‖+ c‖g′‖≤ c‖∆(σ2g′)‖+ c‖σ∇g′‖+ c‖g′‖≤ c‖σ2∆g′‖+ c‖σ∇g′‖+ c‖g′‖≤ c‖σ2∂iδ′‖+ c‖σ∇g′‖+ c‖g′‖≤ c+ c‖σ∇g′‖+ c‖g′‖.


Further, there holds

‖σ∇g′‖2 = (∇(σ2g′),∇g′)− (∇σ2g′,∇g′)= (∂i(σ

2g′), δ′)B∗+ 1

2(∆σ2g′, g′),

= −(σ2g′, ∂iδ′)B∗

+ 2‖g′‖2

≤ cρ−1∗ ‖g′‖1;B∗

+ 2‖g′‖2

≤ c(1 + ‖g′‖2).

Let z ∈ V ∩H2(Ω) be the solution of the auxiliary problem

(∇z,∇ϕ) = (g′, ϕ)‖g′‖−1,

satisfying ‖z‖2,2 ≤ c . With this notation, there holds

‖g′‖2 = (∇z,∇g′) = (∇z, δ′) ≤ cρ−2∗ ‖∇z‖1;B∗

≤ cL(ρ∗)‖z‖2,2 ≤ cL(ρ∗).

This finally implies the estimate (2.1.50). Q.E.D.

2.2 Solution of the discretized problems

In the following, we discuss techniques for solving the nonlinear algebraic system resulting fromthe finite element discretization of a quasi-linear elliptic problem as considered above,

a(uh;ϕh) = (f, ϕh) ∀ϕh ∈ Vh, (2.2.51)

We concentrate on the special case that the nonlinear problem has the form

a(u;ϕ) = (F (∇u),∇ϕ) = (K(∇u)∇u,∇ϕ) ∀ϕ ∈ V.

As particular example, we keep in mind the modified p-Laplace problem with

F (∇u) = K(∇u)∇u =(1 + |∇u|2

)p/2−1∇u, 1 ≤ p <∞.

2.2.1 A brief survey of iterative solution methods

Let ϕih, i = 1, . . . , Nh = dimVh be the usual “nodal basis” of the finite element subspaceVh ⊂ V = H1

0 (Ω) . In the case of “linear” finite elements these basis functiona are defined bythe relation (so-called “Lagrange basis”)

ϕih(aj) = δij , i, j = 1, . . . , Nh, aj interior nodal point of triangulation Th.

Then, using the representation uh =∑Nh

j=1 xjϕjh the finite dimensional problem (2.2.51) can

equivalently be written in the form

a(uh;ϕih) = (f, ϕih) i = 1, . . . , Nh. (2.2.52)

2.2 Solution of the discretized problems 55

This is a nonlinear algebraic system for the coefficient vector x = (xj)Nh

j=1 of the form

A(x)x = b, (2.2.53)

where, analogously to the linear case, the (nonlinear) system matrix A(x) = (a(x)ij)Nh

i,j=1 and

the load vector b = (bi)Nh

i=1 are given by

a(x)ij = (K(∇uh)∇ϕjh,∇ϕih), , bi = (f, ϕih)

For solving this problem, we consider the following standard iterative methods:

1. Fixed-point (defect correction) iterationThe nonlinear system (2.2.53) is reformulated as a fixed point equation

Cx = Cx+ b−A(x)x, C preconditioning matrix,

which is then solved by the corresponding fixed point (defect correction) iteration

Cδxt = dt := b−A(xt−1)xt−1, xt = xt−1 + δxt. (2.2.54)

By the Banach fixed point theorem this iteration converges for appropriate starting valuesx0 if the fixed point mapping

G(x) := (I − C−1A(x))x + C−1b

is a contraction on a closed subset M ∈ RNh . From the linear situation with A(x) = A , we

know that the speed of the convergence of such iterations depends essentially on the choiceof the “preconditioner” C . This ranges from simple diagonal scaling such as in the Jacobiiteration with strong mesh-dependence up to sophisticated nonlinear multigrid schemeswith sometimes almost mesh-independent behavior. In the nonlinear case considered, thebehavior of these methods will not be better than in the linear case. Therefore, we have toexpect a strong mesh-dependence of the convergence speed, i. e., the number of iterationsrequired on a certain mesh Th to reach the level of the discretization error grows rapidlywith the dimension Nh (like h−2 in the simplest case). The defect correction iterationcan also be formulated within the function space framework:

c(δuth, ϕh) = dt(ut−1h ;ϕh) := (f, ϕh)− a(ut−1

h ;ϕh), ∀ϕh ∈ Vh, uth = ut−1h + δuth,

with an suitable (regular) bilinear form c(·, ·) (for preconditioning).

2. Functional iterationProblems of the form (2.2.53) can often be solved by a so-called “functional iteration”,in which starting with a suitable initial guess x0 a sequence of iterates is computed bysuccessively solving the linear equations

A(xt−1)xt = b, t ∈ N. (2.2.55)

This iteration can also be formulated in function space. Starting from a suitable initialguess u0h ∈ Vh one computes iterates uth ∈ Vh by sucessively solving the linear problems

(K(∇ut−1h )∇uth,∇ϕh) = (f, ϕh) ∀ϕh ∈ Vh.


Functional iteration is particularly attractive if the problem is only mildly nonlinear suchas the “semi-linear” diffusion equation

−∇ · (a(u)∇u) = f,

or the diffusion-transport equation

−ν∆u+ β(u) · ∇u = f.

Using the finite dimensionality of the problem one can often establish the convergence ofthe functional iteration, however without estimates for the speed of convergence.

3. Gradient methodIf the nonlinear problem originates from a minimization problem, one may use a so-called“descent method” for its solution. The classical representative of this type of methods isthe “gradient method”. We formulate this method in the abstract Hilbert space setting.Here, starting from a suitable initial guess u0h ∈ Vh one determines a sequence of iterates bysuccessively solving the one-dimensional minimization problems (so-called “line search”)

ut+1h = uth − λtg

th : J(ut+1

h ) = minλ∈R+

J(uth − λgth), (2.2.56)

where the descent direction (negative gradient direction) is determined by the linear equa-tion

(gth, ϕh) = J ′(uth)(ϕh) ∀ϕh ∈ Vh.

Under certain conditions on the functional J(·) this method is known to converge globallywith linear rate depending on the mesh size h .

In the linear case, which corresponds to the quadratic functional

J(uh) =12a(uh, uh)− (f, uh)

the gradient method looks particularly simple. The gradient at a point uth ∈ Vh is thedefect functional

J ′(uth)(ϕh) = a(uth, ϕh)− (f, ϕh), ϕh ∈ Vh,

and the corresponding gradient direction gth ∈ Vh is given as the solution of the equation

(gth, ϕh) = a(uth, ϕh)− (f, ϕh), ϕh ∈ Vh,

which is easy to solve. Then, the optimal step length λt in the line search is determinedby

λt =a(uth, g

th)− (f, gth)

a(gth, gth)

=‖gth‖2a(gth, g

th).

The gradient method in its original form is as slow as the simplest fixed point iterations(Jacobi or Gauss-Seidel method) and its convergence depends strongly on the mesh size.

4. Newton iterationThe most popular method for solving nonlinear problems of type (2.2.53) is the Newtonmethod and its variants. The classical Newton iteration for the system A(x)x = b in


defect correction form reads as follows:

[xtTA′(xt) +A(xt)]δxt = b−A(xt)xt, xt+1 = xt + δxt, (2.2.57)

where A′(x) denotes the derivative of the matrix A(·) . Below, we shall consider the New-ton method at first in R

n . Here, the classical theorem of Newton-Kantorovich ensures thelocal quadratic convergence of the Newton method under certain structural assumptionson the problem considered. However, in this formulation the dependence of the conver-gence on the dimension n , i. e., the mesh size in the present situation, does not becomeclear. Therefore, subsequently, we will consider the Newton method in a general functionspace setting,

a′(uth;ut+1h , vh) = a′(uth;u

t−1h , vh) + (f, vh)− a(uth; vh), ∀vh ∈ Vh, (2.2.58)

or in form of a defect correction process,

a′(uth; δuth, vh) = (f, vh)− a(uth; vh) ∀vh ∈ Vh, ut+1

h = uth + δu7h. (2.2.59)

which allows us to prove convergence almost independent of the mesh size (so-called “meshindependence principle”).

2.2.2 The Newton method in Rn

We give a brief analysis of the classical Newton method in Rn for the approximation of roots of

differentiable functions. Here, the notation ‖ · ‖ is used for any vector norm and the associatednatural matrix norm (e. g., the Euclidean norm). The Euclidean scalar product is denoted by〈·, ·〉 .

Let D ⊂ Rn be an open non-empty set and f : D → R

n a differentiable function for whicha root z ∈ D is to be computed. The Jacobi matrix f ′(·) is assumed to be regular on the levelset

D∗ :=x ∈ D | ‖f(x)‖ ≤ ‖f(x∗)‖

, x∗ ∈ D arbitrarily fixed,

with uniformly bounded inverse,

‖f ′(x)−1‖ ≤ β, x ∈ D∗.

Further let f ′(·) be uniformly Lipschitz continuous on D∗ ,

‖f ′(x)− f ′(y)‖ ≤ γ‖x− y‖, x, y ∈ D∗.

For this finite dimensional setting, we have the following Theorem of Newton-Kantorovich.

Theorem 2.4 (Newton-Kantorovich): Under the above assumptions let for the initial pointx0 ∈ D∗ with α := ‖f ′(x0)−1f(x0)‖ the following condition be satisfies:

q := 12αβγ < 1. (2.2.60)

Then, the Newton iteration

f ′(xt)xt+1 = f ′(xt)xt − f(xt), t ≥ 1, (2.2.61)


generates a sequence (xt)t∈N ⊂ D∗ , which converges quadratically to a root z ∈ D∗ of f .Further, there holds the a priori error estimate

‖xt − z‖ ≤ α

1− qq(2

t−1) , t ≥ 1. (2.2.62)

Proof: To the starting point x0 ∈ D∗ there corresponds the non-empty, closed level set

D0 :=x ∈ D | ‖f(x)‖ ≤ ‖f(x0)‖

⊂ D∗.

We consider the continuous mapping g : D0 → Rd defined by

g(x) := x− f ′(x)−1f(x), x ∈ D0.

(i) First, we derive some auxiliary results. For x ∈ D0 , we set

xr := x− rf ′(x)−1f(x), 0 ≤ r ≤ 1,

and

R := maxr ∈ [0, 1] |xs ∈ D0, 0 ≤ s ≤ r

= max

r ∈ [0, 1] | ‖f(xs)‖ ≤ ‖f(x0)‖, 0 ≤ s ≤ r

.

For the vector function h(r) := f(xr) there holds

h′(r) = −f ′(xr)f ′(x)−1f(x), h′(0) = −h(0).

For 0 ≤ r ≤ R this yields

‖f(xr)‖ − (1−r)‖f(x)‖ ≤ ‖f(xr)− (1−r)f(x)‖ = ‖h(r)− (1−r)h(0)‖

=∥∥∥∫ r

0h′(s) ds+ rh(0)

∥∥∥ =∥∥∥∫ r

0h′(s)− h′(0) ds

∥∥∥

≤∫ r

0‖h′(s)− h′(0)‖ ds,

and further obserbing xs − x = −sf ′(x)−1f(x) :

‖h′(s)− h′(0)‖ = ‖f ′(xs)− f ′(x)f ′(x)−1f(x)‖≤ γ‖xs − x‖‖f ′(x)−1f(x)‖ ≤ γs‖f ′(x)−1f(x)‖2.

This yields

‖f(xr)‖ − (1−r)‖f(x)‖ ≤ 12r

2γ‖f ′(x)−1f(x)‖2. (2.2.63)

With the quantity αx := ‖f ′(x)−1f(x)‖ and the assumption ‖f ′(x)−1‖ ≤ β it follows that

‖f(xr)‖ ≤ (1− r + 12r

2αxβγ)‖f(x)‖.

In case that αx ≤ α , we then obtain in view of the assumption 12αβγ < 1 :

‖f(xr)‖ ≤ (1− r + r2)‖f(x)‖.


Consequently R = 1 in this case, i. e., g(x) ∈ D0 . For such x ∈ D0 , it follows that

‖g(x) − g2(x)‖ = ‖g(x) − g(x) + f ′(g(x))−1f(g(x))‖ ≤ β‖f(g(x))‖.

With the help of the estimate (2.2.63) for r = 1 , we obtain:

‖g(x) − g2(x)‖ ≤ 12βγ‖f ′(x)−1f(x)‖2 = 1

2βγ‖x− g(x)‖2. (2.2.64)

(ii) Next, we show that the Newton iterates (xt)t∈N exist in D0 and satisfy the inequality

‖xt − g(xt)‖ = ‖f ′(xt)−1f(xt)‖ ≤ α.

This is done by an induction argument. For t = 0 the assertion is obviously valid. Particularly,since αx0 = α there holds g(x0) ∈ D0 . Let now xt ∈ D0 be an iterate with g(xt) ∈ D0 and‖xt − g(xt)‖ ≤ α . Then,

‖xt+1 − g(xt+1)‖ = ‖g(xt)− g2(xt)‖ ≤ 12βγ‖xt − g(xt)‖2 ≤ 1

2α2βγ ≤ α

and consequently in virtue of the above results g(xt+1) ∈ D0 . Therefore, (xt)t∈N ⊂ D0 exists.Next, we show that this sequence is a Cauchy sequence. With the help of (2.2.64), we obtain

‖xt+1 − xt‖ = ‖g2(xt−1)− g(xt−1)‖ ≤ 12βγ‖g(xt−1)− xt−1‖2 = 1

2βγ‖xt − xt−1‖2,

and iterating this inequality,

‖xt+1 − xt‖ ≤ 12βγ

(12βγ‖xt−1 − xt−2‖2

)2 ≤ (12βγ)(22−1)‖xt−1 − xt−2‖(22)

≤ (12βγ)(22−1)

(12βγ‖xt−2 − xt−3‖2

)(22)= (12βγ)

(23−1)‖xt−2 − xt−3‖(23).

Continuing this iteration down to t = 0 and recalling q = 12αβγ < 1 yields

‖xt+1 − xt‖ ≤ (12βγ)(2t−1)‖x1 − x0‖(2t) ≤ (12βγ)

(2t−1)α(2t) ≤ αq(2t−1).

For arbitrary m ∈ N it follows that

‖xt+m − xt‖ ≤ ‖xt+m − xt+m−1‖+ · · ·+ ‖xt+2 − xt+1‖+ ‖xt+1 − xt‖≤ αq(2

t+m−1−1) + · · ·+ αq(2t+1−1) + αq(2

t−1)

≤ αq(2t−1)

(q(2

t))(2m−1−1) + · · ·+ q(2

t) + 1

≤ αq(2t−1)

∞∑

j=0

(q(2t))j ≤ αq(2

t−1)

1− q(2t).

This shows that (xt)t∈N ⊂ D0 is actually a Cauchy sequence. Its limit z ∈ D0 is necessarily afixed point of g and a root of f ,

z = limt→∞

xt = limt→∞

g(xt−1) = g(z).


Letting m→ ∞ yields the asserted a priori error estimate (2.2.62),

‖z − xt‖ ≤ αq(2t−1)

1− q(2t)

≤ α

1− qq(2

t−1).

To prove the a posteriori error estimate (??, we note that

f(xt) = f(xt)− f(z) = f ′(ξt)(xt − z), ξt ∈ (xt, z),

what completes the proof. Q.E.D.

Remark 2.7: In the theorem of Newton-Kantorovich as stated above the assumptions are de-signed in such a way that also the existence of a root can be guaranteed. If this existence isbeing made an additional assumption, the argument for proving the local quadratic convergenceof the Newton iteration can be significantly simplified.

In the realization of the Newton method there one has to cope with two main difficulties:

(i) high computational cost in each iteration step,

(ii) sufficiently “good” starting value x0 required.

For overcome the first one of these difficulties one may use the so-called “simplified Newtoniteration”,

f ′(c)δxt = dt := −f(xt), xt+1 = xt + δxt, (2.2.65)

with a suitable point c ∈ Rn , e. g., c = x0 , lying sufficiently close to the root z . In this iteration

all linar systems to be solved have the same coefficient matrix, which can be utilized to lowerthe cost of these substeps (e. g., by computing once an LR decomposition of f ′(c) and usingthis in all subsequent iteration steps). However, this modification reduces the Newton methodto a simple fixed point iteration with an only linear speed of convergence. To lower the difficultyof generating an appropriate starting value, one may try to enlarge the region of convergenceof the Newton method by introducing a “damping” λt ∈ (0, 1] , which is adaptively adjusted inthe course of the iteration,

f ′(xt)δxt = dt := −f(xt), xt+1 = xt + λtδxt. (2.2.66)

The following theorem contains a useful damping strategy.

Theorem 2.5 (Damped Newton method): Let the assumptions of Theorem 2.4 be satisfied.Then following the rule

λt := min1,

1

αtβγ

, αt := ‖f ′(xt)−1f(xt)‖, (2.2.67)

the damped Newton iteration (2.2.66) generates for any starting value x0 ∈ D∗ a sequence(xt)t∈N , for which after t∗ steps the condition q∗ :=

12αt∗βγ < 1 is satisfied. Then, for t ≥ t∗

the iterates xt converge quadratically to a root z of f(x) with the a priori error estimate

‖xt − z‖ ≤ α

1− q∗q(2t−1)∗ , t ≥ t∗. (2.2.68)


Proof: We use the notation from the proof of Theorem 2.4. For some x ∈ D0 there holds,setting again xr := x− rf ′(x)−1f(x), 0 ≤ r ≤ 1 , and αx := ‖f ′(x)−1f(x)‖ the estimate

‖f(xr)‖ ≤ (1− r + 12r

2αxβγ)‖f(x)‖, 0 ≤ r ≤ R = maxr ∈ [0, 1] |xs ∈ D0, 0 ≤ s ≤ r.

The prefactor becoms minimal for

r∗ = min1,

1

αxβγ

> 0 : 1− r∗ +

12r

2∗αxβγ ≤ 1− 1

2αxβγ< 1.

For the choice of

rt := min1,

1

αtβγ

, αt := ‖f ′(xt)−1f(xt)‖ ≤ β‖f(xt)‖,

we have (xt)t∈N ⊂ D0 , and the norm ‖f(xt)‖ is strictly monotonically decreasing, i. e.,

‖f(xt+1)‖ ≤(1− 1

2αtβγ

)‖f(xt)‖.

Therefore, after finitely many, t∗ ≥ 1 , iteration steps, we have 12αt∗βγ < 1 , and the quadratic

convergence of the subsequent iteration (xt)t≥t∗ follows from Theorem 2.4. Q.E.D.

2.2.3 The Newton method in function space

Now. we consider the Newton method formulated in function space for computing the solutionsuh ∈ Vh ∩ BR(u) of the discretized problems corresponding to a general quasi-linear ellipticproblem.

a(u;ϕ) := (F (∇u),∇ϕ) = (f, ϕ) ∀ϕ ∈ V = H10 (Ω).

Suppose that the discretization uses a subspace Vh ⊂ V of piecewise linear elements on a regulartriangulation Th of Ω with the usual nodal basis ϕih, i = 1, . . . , Nh = dimVh . Then, thediscrete problem in the function space Vh

a(uh;ϕh) = (f, ϕh) ∀ϕh ∈ Vh,

is equivalent to a nonlinear algebraic system for the coefficient vector x = (xj)Nh

j=1 ∈ RNh in the

representation uh =∑Nh

j=1 xjϕjh ,

f(x) = 0,

where the mapping f = (fi)Nh

i=1 : D ⊂ RNh → R

Nh is givem by

fi(x) := a(

Nh∑

j=1

xjϕjh;ϕ

ih)− (f, ϕih) = (F (

Nh∑

j=1

xj∇ϕjh),∇ϕih)− (f, ϕih), i = 1, . . . , Nh.

As usual the vector function F (η) = (Fi(η))2i=1 , is assumed to be continuously differentiable

with, for bounded arguments, positive definite and Lipschitz continuous Jacobian matrix F ′(η) =(∂ηjFi(η))

2i,j=1 :

〈F ′(η)ξ, ξ〉 ≥ α|ξ|2, ξ ∈ R2, |F ′(η) − F ′(η′)| ≤ γ|η − η′|, η, η′ ∈ BR.


We want to show that these properties carry over to the mapping f(·) . The Jacobian matrixf ′(·) of the mapping f(·) has the elements

f ′ij(x) =∂

∂xjfi(x) = (F ′(

Nh∑

j=1

xj∇ϕjh)∇ϕjh,∇ϕih), i, l = 1, . . . , Nh.

Hence, for any two finite element functions and their corresponding nodal basis representations,vh =

∑Nh

j=1 yjϕjh, wh =

∑Nh

j=1 zjϕjh ∈ Vh , there holds

〈f ′(x)y, z〉 =Nh∑

i,k=1

〈f ′ik(x)yk, zi〉 =Nh∑

i,k=1

(F ′(

Nh∑

j=1

xj∇ϕjh)yk∇ϕkh, zi∇ϕih)

= (F ′(uh)∇vh,∇wh) = a′(uh; vh, wh).

The usual assumptions on F (·) imply that the tangent form a′(uh; ,·) is uniformly V -ellipticand Lipschitz continuous, for arguments uh, u

′h ∈ Vh ∩BR(u) , in the following sense:

a′(uh; vv , vh) ≥ α‖∇vh‖2,

|a′(uh; vh, wh)− a′(u′h; vh, wh)| ≤ γ‖∇(uh − u′h)‖∞‖∇vh‖‖∇wh‖.

Obviously, these properties carry over to the Jacobian matrix f ′(·) as follows:

(f ′(x)y, y) = a′(uh; vh, vh) ≥ α‖∇vh‖2 ≥ α〈Ay, y〉 = α‖y‖2A, y ∈ RNh ,

|f ′(x; y, z) − f ′(x′; y, z)| = a′(uh; vh, vh)− a′(u′h; vh, vh)

≤ γ‖∇(uh − u′h)‖∞‖∇vh‖‖∇wh‖≤ γ‖∇(uh − u′h)‖∞‖y‖A‖z‖A≤ cγh−1‖x− x′‖A‖y‖A‖z‖A,

with the positive definite (not uniformly in h ) “stiffness matrix” A = ((∇ϕjh,∇ϕih))Nh

i,j=1

In view of the foregoing results, it appears most natural in the present situation to use thetheorem of Newton-Kantorovich for the special mapping f(·) with the norm ‖ · ‖ := ‖ · ‖A =〈A·, ·〉1/2 since then all assumption of the theorem are satisfied. However, due to the use of theinverse relation

‖∇vh‖∞ ≤ ch−1‖∇vh‖, vh ∈ Vh,the Lipschitz continuity of the Jacobian f ′(·) is strongly h-dependent with a constant behavinglike O(h−1) . On the other hand, the direct use of the norm ‖ · ‖ := ‖∇ · ‖∞ leads to likewiseinpractical bounds for the inverse of the Jacobian matrix f ′(x)−1 . This incompatibility results inunrealistically strong requirements on the quality of the initial guess x0 to guarantee convergencewithin the BR-ball of the solution u . For this reason, in our further analysis of the Newtonmethod, we prefer the function space setting, which is more appropriate for the structuralproperties of the nonlinear problems considered.

Starting with a suitable initial guess u0h ∈ Vh ∩ BR(u) , one obtains a sequence of iteratesuth ∈ Vh ∩BR(u), t ∈ N, by sucessively solving the linear problems

a′(uth;ut+1h , vh) = a′(uth;u

t−1h , vh)− a(uth; vh) + (f, vh), ∀vh ∈ Vh, (2.2.69)


or in form of a defect correction process,

a′(uth; δuth, vh) = (f, vh)− a(uth; vh) ∀vh ∈ Vh, ut+1

h = uth + δu7h. (2.2.70)

In view of Theorem 2.1, the theorem of Newton-Kantorovich, and the foregoing discussion thelocal quadratic convergence of this iteration is guaranteed for any fixed h sufficiently small,say h ≤ h1 ≤ h0 , if the starting value u0h is taken sufficiently close to the discrete solutionuh ∈ Vh ∩ BR(u) . But, since our assumptions do not ensure the convergence of the Newtoniteration in function space for the “continuous” problem, it cannot be expected that the speedof convergence is uniform as h → 0 . However, even in the present rather general situation,we have the following almost mesh-independence result for the Newton method. The proof isless complicated than that of the theorem of Newton-Kantorovic since the existence of a root ofthe nonlinear mapping considered is a priori known and does not need to be established in thecourse of the argument.

Theorem 2.6 (Mesh-independence of Newton method): For sufficiently small h, h ≤h1 , there exist positive constants c1, c2, c3 independent of h , such that for any starting valueu0h ∈ Vh ∩BR(u) satisfying

|||u− u0h|||∞ ≤ c1L(h)−1, (2.2.71)

the Newton iterates uth ∈ Vh∩BR(u), t ∈ N, are well defined and the accuray of the discretizationerror is reached after at most th ≤ c2L(L(h)) steps,

|||u− uth|||∞ ≤ c3L(h)h1−2/p, t ≥ th. (2.2.72)

Proof: The proof is similar to that of Theorem 2.1.

i) First, assuming the existence of the iterate ut−1h ∈ Vh ∩BR(u) , for some t ∈ N , we consider

the next iterate uth ∈ Vh , which is defined by

a′(ut−1h ;uth, vh) = a′(ut−1

h ;ut−1h , vh)− a(ut−1

h ; vh) + (f, vh), ∀vh ∈ Vh.

For this, we derive the identity

a′(u;Rhu− uth, vh) = a′(u;u− uth, vh)− a′(ut−1h ;u, vh) + a′(ut−1

h ;u, vh)

= a′(u;u− uth, vh)− a′(ut−1h ;u− uth, vh) + a′(ut−1

h ;u− ut−1h , vh)

− a(ut−1h ; vh) + (f, vh)



− a(ut−1h ; vh) + a(u; vh)



−∫ 1

0a′(ut−1

h + s(u− ut−1h );u− ut−1

h , vh) ds

= a′(u;u− uth, vh)− a′(ut−1h ;u− uth, vh)

+

∫ 1

0

a′(ut−1

h ;u− ut−1h , vh)− a′(ut−1

h + s(u− ut−1h );u− ut−1

h , vh)ds.

The right-hand side is again viewed as a functional v∗h ∈ V ∗h acting on the function vh ∈ Vh ,


with dual norm

‖v∗h‖∞;h ≤ c(R)|||u− uth|||∞ + |||u− ut−1

h |||∞|||u− ut−1

h |||∞.

Then, analogously as in the proof of Theorem 2.1, we conclude the estimate

|||u− uth|||∞ ≤ c′L(h)h1−2/p + c′′L(h)|||u − uth|||∞ + |||u− ut−1h |||∞|||u − ut−1

h |||∞, (2.2.73)

with constants c′, c′′ independent of h and t . Setting

at := |||u− uth|||∞, δ := c′L(h)h1−2/p, β := c′′L(h),

this readsat ≤ δ + β(at + at−1)at−1, t ∈ N.

ii) We want to show that, for sufficiently small h ,

at ≤ 3δ +1

3β(3βa0)

2t , t ∈ N. (2.2.74)

From this, one easily obtains that, for a starting value u0h satisfying (2.2.71), all iterates uth ∈Vh ∩BR(u) exist and satisfy (2.2.72) after t ≤ th = c2L(L(h)) steps,

|||u− uth|||∞ = at ≤ 3δ +1

3β(3βa0)

2t

≤ 3c′L(h)h1−2/p +1

3β(3c′′L(h)c1L(h)

−1)2t

≤ 3c′L(h)h1−2/p +1

3β(3c′′c1)

2t

≤ 6c′L(h)h1−2/p,

for 3c′′c1 =: q < 1 and t ≥ t∗ ≈ L(L(h)) determined by

q2t ≤ 3c′L(h)h1−2/p, 2t log(q) ≤ log(3c′L(h)h1−2/p) ≈ L(h), t log(2) ≈ L(L(h)).

iii) To prove (2.2.74), we set bt := 3βat and require h to be sufficiently small, such that

c′c′′L(h)2h1−2/p ≤ 19 .

Then, we have

bt = 3βat ≤ 3βδ + 13 (3βat + 3βat−1)3βat−1 = 3βδ + 1

3(bt + bt−1)bt−1,

and further, in view of the above assumptions,

bt ≤ 3c′c′′L(h)2h1−2/p + 13(bt + bt−1)bt−1 ≤ 1

3 +13(bt + bt−1)bt−1.

By assumption, for c1 ≤ 1/3 , there holds b0 = 3βa0 ≤ 3L(h)c1L(h)−1 ≤ 1 . Then, by induction,

we find that bt ≤ 1, t ∈ N , and, consequently,

bt ≤ 92βδ +

12b

2t−1.


From this, we infer by induction that

bt ≤ 9βδ + b2t

0 , t ≥ 1.

which implies (2.2.74). Obviously, this inequality holds true for t = 1 . Suppose that it holdstrue for some t− 1 ≥ 1 . Then,

bt ≤ 92βδ +

12b

2t−1 ≤ 9

2βδ +12

(9βδ + b2

t−1

0

)2

≤ 92βδ + (9βδ)2 + b2

t

0 ≤ 9βδ + b2t

0 ,

for h sufficiently small such that 9βδ ≤ 1/2 . This complestes the proof. Q.E.D.

2.2.4 The projective Newton method

Next, we consider a multi-level variant of the Newton method (2.2.69), which works on suces-sively refined meshes and is sometimes referred to as “projective Newton method”. Let againVt := Vht ⊂ V be the usual low-order finite element subspaces with decreasing mesh sizesh0 > h1 > · · · > ht → 0 (t → ∞) . Starting with an initial guess u0 ∈ V0 on the coarsest mesh,one determines increasingly accurate approximations ut ∈ Vt, t ∈ N, by sucessively solving thelinear problems

a′(ut−1;ut, v) = a′(ut−1;ut−1, v)− a(ut−1; v) + (f, v) ∀v ∈ Vt. (2.2.75)

These problems are of increasing complexity, i. e., dimVt → ∞ (t → ∞) . The question is nowhow rapidly the mesh size ht may be decreased in this process without loosing the convergenceproperties of the Newton method. The following theorem gives an answer to this question.

Theorem 2.7 (Projective Newton method): Suppose that the coarsest mesh size h0 is suf-ficiently small and that the initial guess u0 ∈ V0 ∩BR(u) satisfies

|||u− u0|||∞ ≤ c0h1−2/p0 . (2.2.76)

Further, let the mesh sizes ht be chosen such that

ht ≥ κh2t−1L(ht−1)p/(p−2), (2.2.77)

with a sufficiently large constant κ > 0 . Then, the iterates ut ∈ Vt ∩ BR(u) are well definedand there holds

|||u− ut|||∞ ≤ c1h1−2/pt . (2.2.78)

Proof: The proof follws the same line of argument as already used in the proof of Theorem 2.6.Assuming that the iterate ut−1 ∈ Vt−1 ∩BR(u) exists, we obtain the following estimate for thenext iterate ut ∈ Vt :

|||u− ut|||∞ ≤ c′h1−2/pt + c′′L(ht)

|||u− ut|||∞ + |||u− ut−1|||∞

|||u− ut−1|||∞ .


Setting at := |||u− ut|||∞ , δt := c′h1−2/pt , and βt := c′′L(ht) this inequality reads

at ≤ δt + βt(at + at−1)at−1.

Next, we set σ := supt∈N βt/βt−1 ≥ 1 and fix a number m ≥ 2σ + 1 . Then for h0 sufficientlysmall, such that a0 ≤ (mβ1)

−1 and δt ≤ (m2βt)−1 , an elemantary induction argument shows

that at ≤ (mσβt)−1 . This implies that

at ≤ γδt + γβta2t−1, (2.2.79)

where γ := m/(m − 1) . To show the error estimate (2.2.78), we note that it clearly holds fort = 1 with c1 := 2γσ . Suppose that it holds true for some t− 1 ≥ 2 . Then,

at ≤ γδt + γβta2t−1 ≤ γδt + κ−2γc′′c21

(L(ht)

p/(p−2)h2t−1

)1−2/p,

and, consequently, at ≤ c1δt , for sufficiently large κ . From this the asserted estimate followsagain by induction. Q.E.D.

2.3 Exercises

Exercise 2.1: Let Ω ⊂ R2 be a convex polygonal domain, V := H1

0 (Ω) and Vh ⊂ V the usualapproximating finite element spaces consisting of piecewise linear elements on a quasi-uniformfamily Thh>0 of triangulations. Derive for the Ritz projection Rh : V → Vh , defined by

(∇Rhv,∇ϕ) = (∇v,∇ϕ) ∀ϕ ∈ V,

the L2-stability estimate

‖Rhv‖ ≤ c‖v‖ + ch‖∇v‖, v ∈ V ∩H2(Ω),

and use this to prove the optimal-order L2-error estimate

‖u−Rhu‖ ≤ ch2‖u‖2,2, v ∈ V ∩H2(Ω).

Do these results also hold true on meshes not necessarily satisfying the uniform size condition?(Hint: Use the standard “duality argument”.)

Exercise 2.2: Show that the function

u(x) = (x1 + x2) log(|x|), x ∈ Ω := y ∈ R2 | |y| < 1,

is in V = H10 (Ω) and solves the variational equation

(∇u,∇ϕ)Ω = (g,∇ϕ)Ω ∀ϕ ∈ V,

where the right-hand side vector g = (g1, g2) is given as

g(x) = |x|−2(x21 + 2x1x2 − x22,−x21 + 2x1x2 + x22)T .

The right-hand side g is bounded while ∇u has a logarithmic singularity. This demonstrates

2.3 Exercises 67

that the log(1/h) in the stability estimate

|||Rhg|||∞ ≤ c log(1/h)‖g‖∞;h,

for the extended Ritz projection Rh : V ∗h → Vh introduced in class cannot be removed.

Exercise 2.3: With the notation introduced in class consider the standard low-order finiteelement approximation of the quasi-linear boundary value problem

−∇ · F (·,∇u) = f in Ω, u = g on ∂Ω,

on a convex polygonal domain Ω ⊂ R2 with smooth boundary data g ∈ H2,p(Ω) for some

p > 2 . Suppose that there exists a solution u ∈ H2,p(Ω) . Then, the main theorem presentedin class for homogeneous boundary data also applies to this more general situation. Use this toshow for the minimal surface problem the optimal-order L2-error estimate

‖u− uh‖ ≤ c(u)h2.

(Hint: Check that in the minimal surface problem the assumptions made for the main theoremare satisfied.)

Exercise 2.4: In the proof of the Main Theorem on the finite element approximation of generalquasi-linear elliptic problems, we have made use of the fact that a subset Θh of the interval[0, 1] , which is nonempty and open as well as closed with respect to [0, 1] necessarily coincideswith the whole interval, i. e., Θh = [0, 1] . Give an argument for this fact.

Exercise 2.5: Extend the proof of the Main Theorem on the finite element approximation ofgeneral quasi-linear elliptic problems given in class, more precisely the homotopy argument, tocover also the H1-error estimate

‖u− uh‖1,2 ≤ c(u)h,

for solutions u ∈ V = H10 (Ω) satisfying u ∈ V ∩H2,p(Ω) for some p > 2. To this end, one may

use the following stability estimate for the Ritz projection Rh :

‖∇Rhv∗h‖ ≤ c‖v∗h‖2;h, ‖v∗h‖2;h := supv∗h(vh), ‖∇vh‖ = 1.

(Hint: Define an appropriate set Θh ⊂ [0, 1] and show that it coincides with [0, 1] . In provingthe closedness of Θh , among others, the above stability estimate is needed.)

Exercise 2.6: Let Thh>0 be a family of triangulations in Rd, d = 2, 3, which is strongly

shape uniform but needs to be only weakly size uniform. For any cell T ∈ Th let BT ⊂ Tdenote the maximal inscribed circle. Derive for piecewise linear elements the estimate

‖uh‖∞;T ≤ c‖uh‖∞;BT, uh|T ∈ P1(T ),

with a constant c independent of T and h . Can this estimate also be guaranteed if the familyof meshes is not strongly shape uniform? (Hint: One possible argument is based on Taylorexpansion.)


Exercise 2.7: In class the “real” Green’s function Gx(y) has been used, for which in 2D, weknow the estimate

|Gx(y)| ≤ c| log(|x− y|)|+ 1.

Use this to show that on any circle B(x) with midpoint x and radius ρ ≤ 1 , there holds theestimate

|B(x)|−1

∫

B(x)|Gx(y)| dy ≤ cL(ρ),

with a constant c independent of x and ρ . (Hint: Rewrite the integral using polar coordinates.)

Exercise 2.8: In class the L∞-stability estimate

‖Rhu‖∞ ≤ c‖u||∞ + hL(h)‖∇u‖∞, u ∈ V ∩W 1,∞(Ω),

for the standard Ritz projection Rh : V → Vh has been proven for strongly shape uniform butonly weakly size uniform mesh families. Use this result on strongly size and shape uniform (i.e., quasi-uniform) mesh families together with the inverse relation

‖∇vh‖∞ ≤ ch−1min‖vh‖∞, vh ∈ Vh,

and the interpolation estimate

‖v − Ihv‖∞ + hmax‖∇(v − Ihv)‖∞ ≤ chmax‖∇v‖∞, v ∈W 1,∞(Ω),

to derive the W 1,∞-stability estimate

‖∇Rhu‖∞ ≤ cL(h)‖∇u‖∞, u ∈ V ∩W 1,∞.

Remark: Notice that this argument essentially depends on the strong size uniformity of themeshes and does not work if only weak size uniformity holds.

Exercise 2.9: Let T ∈ Th be a triangle in a shape-regular family of triangulations of a polyg-onal domain Ω ⊂ R

2 . Further, let BT be a maximal inscribed circle of T with midpoint xTand radius ρT . Prove for functions v ∈ H1(Ω) the Sobolev-type inequality

|BT |−1

∫

BT

|v| dx ≤ cL(ρT )‖v‖1,2,

where again L(ρ) := maxlog(1/ρT ), 1 . (Hint: You may use the triangle T containing BT ,polar coordinates, integration by parts and a suitable trace inequality.)

Exercise 2.10: The W 1,∞-stability estimate for the finite element Ritz projection

‖∇Rhu‖∞ ≤ cL(h)‖∇u‖∞,

proven in class involves the logarithmic term L(h) = max| log(1/h)|, 1 . Develop an idea foravoiding the occurrence of this term in the proof in the case that the mesh family considered isquasi-uniform, particularly strongly size-uniform.Remark: This idea cannot work in the proof of the L∞-stability estimate

‖Rhu‖∞ ≤ ‖u‖∞ + cL(h)‖∇u‖∞,

2.3 Exercises 69

since here the logarithmic term is known to be present in general.

Exercise 2.11: State the finite dimensional problems resulting from the usual finite elementdiscretization of the following nonlinear boundary value problems and formulate the Newtoniteration in function space for their solution:

a) Minimal surface problem

min

∫

Ω

√1 + |∇u|2 dx on Vg := v ∈ H1(Ω) | v = g on ∂Ω,

with g the trace of a prescribed function in H1(Ω), Ω ⊂ R2 ,

b) Semi-linear diffusion problem

−∇ · (a(u)∇u) = f in Ω, u = 0 on ∂Ω,

for a scalar function u ∈ V := H10 (Ω), Ω ∈ R

2 , with a continuously differentiable, positivecoefficient function a(·) > 0 ,

c) Diffusion-transport problem (“vector Burgers equation”)

−ν∆u+ u · ∇u = f in Ω, u = 0 on ∂Ω,

for a vector function u ∈ V := H10 (Ω)

2, Ω ⊂ R2, and a prescribed f ∈ L2(Ω)2.

Exercise 2.12: Consider the theorem of Newton-Kantorovich on the convergence of the Newtonmethod in R

n as presented in class. Show that in the one-dimensional case (n=1), under thecorresponding assumptions of this theorem, the following a posteriori error estimate holds:

|xt − z| ≤ β|f(xt)| ≤ 12βγ|xt − xt−1|2, t ≥ 1.

Does the proof of this estimate also work in the multidimensional case n ≥ 2 ?

Exercise 2.13: Let Vh ⊂ V = H10 (Ω) be the usual finite element subspaces of piecewise linear

elements on a quasi-uniform family of triangulations Thh>0 . These finite dimensional spacesmay be equipped with various norms, which for any fixed h are all equivalent, but with h-dependent equivalence constants c(h) . Determine this h-dependence for the following norms:

a) ‖vh‖2 ≤ c1(h)‖vh‖∞ ≤ c2(h)‖vh‖2, vh ∈ Vh,b) ‖vh‖1,2 ≤ c1(h)‖vh‖2 ≤ c2(h)‖vh‖1,2, vh ∈ Vh,

c) ‖vh‖2 ≤ c1(h)‖∇vh‖∞ ≤ c2(h)‖vh‖2, vh ∈ Vh.

Exercise 2.14: Consider the finite element discretization of a quasi-linear elliptic boundaryvalue problem on a convex polygonal domain Ω ⊂ R

2 ,

a(u;ϕ) := (F (∇u),∇ϕ) = (f, ϕ) ∀ϕ ∈ V = H10 (Ω).

The discretization uses subspaces Vh ⊂ V of piecewise linear elements on quasi-uniform trian-gulation Th of Ω with the usual nodal bases ϕih, i = 1, . . . , Nh = dimVh . Then, the discrete


problems in the function spaces Vh

a(uh;ϕh) = (f, ϕh) ∀ϕh ∈ Vh,

are equivalent to nonlinear algebraic systems

f(x) = 0,

for the coefficient vectors x = (xi)Nh

i=1 ∈ RNh in the representations uh =

∑Nh

i=1 xiϕih .

a) Let the vector function F (·) be given in the (linear)( form F (∇u) = a∇u with a constanta > 0 . With the given notation, state explicitly the corresponding equivalent discrete problemsin function space and in R

n .

b) Proceed as in (a) but for the nonlinear vector function

F (∇u) = ∇u√1 + |∇u|2

.

Exercise 2.15: Consider the Newton method for solving the finite dimensional problems re-sulting from the usual finite element discretization of the quasi-linear elliptic boundary valueproblem

a(u;ϕ) := (F (∇u),∇ϕ) = (f, ϕ) ∀ϕ ∈ V = H10 (Ω).

It follows from the theorem of Newton-Kantorovich proven in class that, if the vector functionF (·) is differentiable with positive definite and Lipschitz continuous Jacobian, for fixed h theNewton iteration defined by

a′(ut−1h ;uth, ϕh) = a′(ut−1

h ;ut−1h , ϕh) + (f, ϕh)− .a(ut−1

h ;ϕh) ∀ϕ∈Vh,

converges in the norm |||·|||∞ := ‖∇·‖∞ (locally) quadratically to the discrete solution uh . Usea variant of the argument given in class to show that, if the Jacobian of F (·) is only continuous(not necessarily Lipschitz continuous), the Newton iteration still converges with superlinearspeed,

|||uth − uh|||∞|||ut−1

h − uh|||∞→ 0 (t → ∞).

(Hint: One may base the argument on the proof of the Newton-Kantorovich theorem as givenin class.)

3 The (stationary) Navier-Stokes System

In this chapter, we consider the finite element approximation of the basic problem in mathemati-cal Fluid Mechanics, the so-called “Navier-Stokes equations”, which describe the behavior of theflow of certain fluids or gases. These equations are of diffusion-transport type and in contrastto most models in Structural Mechanics do not originate from a minimization principle. Theirnonlinearity is of very special kind, which allows for a rather complete mathematical treatmentwith respect to theory as well as numerical approximation. The material of this chapter andfurther details can largely be found in the lecture notes Rannacher [4], the books of Temam [19],Galdi [7], Girault&Raviart [27], Brenner&Scott [34], and the articles Heywood&Rannacher[42], Heywood, Rannacher&Turek [43], Rannacher&Turek [54], Rannacher [49, 50, 51].

3.1 The stationary Navier-Stokes equations

With the so-called “kinematic viscosity” parameter ν , we obtain the classical (incompressible)“Navier-Stokes equations” in R

d (d = 2, 3) :

∂tv + v · ∇v − ν∆v +∇p = f, ∇ · v = 0, , (3.1.1)

for the velocity vector v and the scalar pressure p in a viscous fluid under the action of anexterior body force (e. g., gravity) with density f . This system represents the conservationequations for mass and momentum for a homogeneous (uniform material properties), isothermal(constant tempereature), incompressible (constant density set to one) Newtonian (linear ma-terial behavior, i. e. stress-strain relation) fluid. We consider this model on bounded domainsΩ ⊂ R

d (d = 2, 3) , which are assumed to be “sufficiently” regular. The equations (3.1.1) aresupplemented by the “no-slip” condition along rigid walls Γrigid (justified by experimental obser-vation) and non homogeneous Dirichlet conditions at inlets Γin and occasionally Neumann-typeconditions at outlets Γout ( ∂Ω = Γrigid ∪ Γin ∪ Γout ):

v = 0 on Γrigid, v = vin on Γin, ν∂nv + pn = 0 on Γout.

We always assume that the rigid part of the boundary is non-trivial, meas(Γrigid) > 0 . Thereare no boundary conditions explicitly imposed on the pressure. This setting is appropriate formodeling, e. g., flow through a finite pipe around an obstacle, where the flow is driven by a left-hand inflow with prescribed profile along ∂Ωin of a globally defined divergence-free velocity field(e. g. Poiseuille flow) and at the right-hand outlet far behind the obstacle the flow is assumed tobe parallel. The determination of the “correct” boundary condition along the artificial part ofthe boundary, Γout , is a difficult problem depending on the particular configuration considered.In the following, we often assume for simplicity that ∂Ω = Γrigid , i. e., the flow is confined to aclosed box and is solely driven by an exterior body force f .

In the case of large viscosity ν (as typical in biological fluids) the nonlinear acelleerationterm v·∇v can be neglected and the Navier-Stokes system reduces to the linear “Stokes system”:

∂tv − ν∆v +∇p = f, ∇ · v = 0, , (3.1.2)

which looks like the vector-heat equation but considered on the manifold of “incompressible”(“solenoidal” or “divergence-free”) vector-fields. Despite its relatively simple structure, thegeneral nonstationary Navier-Stokes equations pose some still unsolved mathematical questions

71


concerning existence and uniqueness of physical meaningful (i. e., regular and stable) solutions.Another prototypical model situation is that of the “lid-driven cavity” (flow in a closed boxdriven by the movement of the upper part of the wall. In the following, we will restrict us tothe stationary case, i. e., neglect the time derivative in (3.1.1):

−ν∆v + v · ∇v +∇p = f, ∇ · v = 0 . (3.1.3)

For setting up a variational formulation of the (stationary) Navier-Stokes problem, we introducethe following function spaces:

H = H10 (Ω; ΓD)

d := v ∈ H1(Ω)d| v = 0 in ΓD , L = L20(Ω) := q ∈ L2(Ω)| (q, 1)Ω = 0,

with the notation ΓD := Γrigid ∪ Γin . Then, the variational formulation of (3.1.1) reads: Findv ∈ vin +H and p ∈ L , such that

ν(∇v,∇ϕ) + (v · ∇v, ϕ)− (p,∇ · v) = (f, ϕ) ∀ϕ ∈ H, (3.1.4)

(∇ · v, χ) = 0 ∀χ ∈ L. (3.1.5)

Let’s assume that this problem has a solution, which is sufficiently regular for being a “classical”solution. By integration by parts in (3.1.4) and use of the boundary condition ϕ|ΓD

= 0 oneobtains that

∫

Ω−ν∆v + v · ∇v +∇p− fϕdx+

∫

Γout

ν∂nv − pnϕdo = 0, ϕ ∈ H.

Consequently, each sufficiently regular solution of the variational problem is a classical solutionof the Navier-Stokes equation and vice versa and it is obviously divergence-free. Further, itsatisfies the following “natural” boundary condition on Γout :

ν∂nv − pn = 0 on Γout. (3.1.6)

This boundary condition of Neumann-type results automatically from the variational formula-tion of the problem by imposing no explicit condition on Γout , which suggests the name “donothing” outflow boundary condition. This (artificial) boundary condition appears natural aslong as the flow can be assumed to be “parallel” across Γout and no further information aboutthe flow behavior beyond Γout is available. Especially, (3.1.6) is satisfied by Poiseuille flow, ifthe pressure p is set to zero on Γout . Actually, in virtue of the divergence condition, ∇· v = 0 ,it turns out that the boundary condition (3.1.6) additionally implies a Dirichlet-type boundarycondition for the pressure separately on each outlet, i. e., on each component of the outflowboundary,

∫

Γout

p do = 0. (3.1.7)

In the Navier-Stokes equations the pressure p appears under the gradient so that an additional(scalar) condition is required for guaranteeing its uniqueness. In configurations with “free”outflow boundary Γout 6= ∅ this is accomplished by the additional implicit condition (3.1.7). Inthe case Γout = ∅ , instead one requires that p has mean value zero, i. e., p ∈ L2

0(Ω) . Below,we will show that the variational Navier-Stokes problem in the case ∂Ω = ΓD always possessesa (not necessarily unique) solution. This proof requires some preparations.

3.1 The stationary Navier-Stokes equations 73

3.1.1 The Stokes operator

We consider the linear Stokes problem with normalized viscosity parameter ν = 1 and homo-geneous Dirichlet boundary conditions along the whole boundary ( v|∂Ω = 0). Correspondingly,we shall use the function spaces H := H1

0 (Ω)d and L := L2

0(Ω) . Then, the Stokes problemreads: Find v, p ∈ H × L ,such that

(∇v,∇ϕ)− (p,∇ · ϕ) = (f, ϕ) ∀ϕ ∈ H, (3.1.8)

(∇ · v, χ) = 0 ∀χ ∈ L. (3.1.9)

We introduce the further spaces:

J1(Ω) := v ∈ H10 (Ω)

d | ∇ · v = 0 a. e. in Ω,J0(Ω) := v ∈ L2(Ω)d | ∇ · v = 0, n · v|∂Ω = 0 in the “weak” sense.

The space J1(Ω) is eqipped with the scalar product (∇·,∇·) a Hilbert space, and thereforecomplerte. The subspace J0(Ω) ⊂ L2(Ω)d is closed. The orthogonal projection from L2(Ω)d

onto J0(Ω) is denoted by P . The space Φ := ϕ ∈ C∞0 (Ω)d| ∇ · ϕ ≡ 0 of solenoidal “test

functions” is densely contained in J0(Ω) as well as in J1(Ω) . With this notation problem (3.1.8)can be written in the following compact form: Find v ∈ J1(Ω) , such that

(∇v,∇ϕ) = (Pf, ϕ) ∀ϕ ∈ J1(Ω). (3.1.10)

For f ∈ L2(Ω)d , by the Poincare inequality the right-hand side in (3.1.10) defines a boundedlinear functional on J1(Ω) . Hence, by the representation theorem of Riesz there exits a uniquesolution v ∈ J1(Ω) of (3.1.10). Then, the equation (3.1.10) defines a linear operator

S : D(S) ⊂ J1(Ω) ⊂ J0(Ω) → J0(Ω),

the so-called “Stokes operator”. As operator in J0(Ω) it has the representation S = −P∆ andis obviously symmetric and positive definite. Further it is onto and consequently “self-adjoint”.Because of the compactness of the embeddings H1(Ω) → L2(Ω) and J1(Ω) → J0(Ω) the inverseS−1 : J0(Ω) → J0(Ω) is a compact operator. By the general spectral theory of (positive definite)self-adjoint operators with compact inverses in Hilbert spaces, we know that the spectrum (setof singular values) of S consists of real (positive) eigenvalues with finite multiplicities and nofinite accumulstion point:

0 < λ1 ≤ ... ≤ λk ≤ ... .

Further, there exists a corresponding system of L2-orthonormal eigenfunctions wkk∈N , whichis complete in J0(Ω) as well as in J1(Ω) , i. e.: Each v ∈ J0(Ω) possesses an expansion of theform

v =

∞∑

k=1

αkwk, αk = (v,wk).

We will use these properties in the existence proof for solutions of the Navier-Stokes problem,below.


3.1.2 Existence result for the Navier-Stokes problem

We consider again the case of pure Dirichlet boundary conditions: H := H10 (Ω; ΓD)

d andL := L2

0(Ω) . The Dirichlet data along inflow/outflow parts of the boundary are assumed to begiven as traces of a solenoidal vector field vin ∈ H1(Ω)d satisfying the no-slip condition alongΓrigid . Using the function space J1(Ω) the stationary Navier-Stokes problem (3.1.4) is writtenin compact form with pressure being eliminated: Find v ∈ vin + J1(Ω) , such that

ν(∇v,∇ϕ) + (v · ∇v, ϕ) = (f, ϕ) ∀ϕ ∈ J1(Ω). (3.1.11)

This is the starting point for the proof of existence of solutions to the Navier-Stokes equations.For simplicity, from now on we will use the notation V := J1(Ω) for the “solution space”.

The special structure of the nonlinearity in the Navier-Stokes equations is decisive for theiranalysis. For functions u, v, w ∈ V = J1(Ω) there holds

(u · ∇v,w) = (v,∇(vw)) − (u · ∇w, v) = −(∇ · u, vw) − (u · ∇w, v) = −(u · ∇w, v).

Setting here v = w , we obtain the important identity

(u · ∇v, v) = 0 . (3.1.12)

For the precise description of the natural regularity of the right-hand side, we use the “negative”Sobolev norm

‖f‖−1 := supϕ∈H

〈f, ϕ〉‖∇ϕ‖ ,

which is the natural norm of the dual space H−1(Ω) of H . Clearly, for functions f ∈ L2(Ω)d

there holds‖f‖−1 ≤ κ‖f‖,

with the constant κ > 0 in the Poincare inequality

‖ϕ‖ ≤ κ‖∇ϕ‖, ϕ ∈ H.

Theorem 3.1 (Existence theorem): In the case ∂Ω = Γrigid the stationary Navier-Stokesproblem (3.1.4) possesses for any value of the viscosity parameter ν > 0 at least one solutionv, p ∈ H × L . For sufficiently small data c2∗ν

−2‖f‖−1 < 1 this solution is unique.

Proof: i) Existenz: We use the technic of “Galerkin approximation”. With the eigenfunctions ofthe Stokes operator wii∈N , we define the finite dimensional subspaces Vm := span〈w1, ..., wm〉 ⊂V . Problem (3.1.4) is then approximated by the following finite dimensional problems: Findvm ∈ Vm , such that

ν(∇vm,∇ϕ) + (vm · ∇vm, ϕ) = (f, ϕ) ∀ϕ ∈ Vm. (3.1.13)

We want to show that these finite dimensional (nonlinear) problems possess solutions, whichare uniformly bounded in V . Then, by a compactness argument, we conclude the existenceof a solution of the infinite dimensional problem. To each v ∈ Vm , we associate an elementQm(v) ∈ Vm as the solution of the (finite dimensional) linear problem

ν(∇Qm(v),∇ϕ) + (v · ∇Qm(v), ϕ) = (f, ϕ) ∀ϕ ∈ Vm.


The solvability of this linear problem follows from the fact that corresponding homogeneousproblem

ν(∇w,∇ϕ) + (v · ∇w,ϕ) = 0 ∀ϕ ∈ Vmonly has the trivial solution w = 0 , what is easily seen by taking ϕ = w ,

0 = ν‖∇w‖2 + (v · ∇w,w) = ν‖∇w‖2.

This defines a (nonlinear) mapping Qm : Vm → Vm . Setting ϕ := Qm(v) in the definingequation, we get

ν‖∇Qm(v)‖2 = (f,Qm(v)) ≤ ‖f‖−1‖∇Qm(v)‖and, consequently,

‖∇Qm(v)‖ ≤ ‖f‖−1

ν=: R.

Hence, the mapping Qm maps the (compact) ball Vm ∩ BR := v ∈ Vm| | ‖∇v‖ ≤ R intoitself. It is continuous (actually Lipschitz continuous), what can be deduced from the followingrelation for arbitrary v,w ∈ Vm ∩BR :

0 = ν(∇[Qm(v)−Qm(w)],∇ϕ) + (v · ∇Qm(v)− w · ∇Qm(w), ϕ)= ν(∇[Qm(v)−Qm(w)],∇ϕ) + ((v − w) · ∇Qm(v), ϕ)+ (w · ∇[Qm(v) −Qm(w)], ϕ) ∀ϕ ∈ Vm.

In d = 2 and d = 3 dimensions, we have the inequalities (with ‖ · ‖p := ‖ · ‖Lp )

‖w‖3 ≤ c∗‖∇w‖, ‖ϕ‖6 ≤ c∗‖∇ϕ‖, w, ϕ ∈ V,

with a generic constant c∗ only depending on Ω . Hence, chosing ϕ := Qm(v)−Qm(w) yields

ν‖∇[Qm(v)−Qm(w)]‖2 = −((v − w) · ∇Qm(v), Qm(v)−Qm(w))

≤ ‖v − w‖3‖∇Qm(v)‖‖Qm(v)−Qm(w)‖6≤ c2∗‖∇(v −w)‖‖∇Qm(v)‖‖∇(Qm(v)−Qm(w))‖

and further‖∇[Qm(v)−Qm(w)]‖ ≤ c2∗R‖∇(v − w)‖.

Then, by the fixed point theorem of Brouwer the continuous mapping Qm : Vm∩BR → Vm∩BRpossesses (at least) one fixed point vm ∈ Vm . By definition, this fixed point satisfies the equation

ν(∇vm,∇ϕ) + (vm · ∇vm, ϕ) = (f, ϕ) ∀ϕ ∈ Vm,

and the uniform bound ‖∇vm‖ ≤ R . The Hilbert space V ⊂ H is compactly embedded intoJ0(Ω) ⊂ L . Hence, there exists a subsequence (vm′)m′∈N , which converges weakly in J1(Ω)and strongly in J0(Ω) to a function v ∈ J1(Ω) ,

(∇(vm′ − v),∇ϕ) → 0 ∀ϕ ∈ J1(Ω), ‖vm′ − v‖ → 0 (m′ → ∞).

This limit v ∈ J1(Ω) is then also solution of equation (3.1.4). To see this, we take an arbitraryϕ ∈ J1(Ω) und a sequence ϕm′ ∈ Vm′ , such that ‖∇(ϕ − ϕm′)‖ → 0 (m′ → ∞) . Then, for


m′ → ∞ (exercise),

(∇vm′ ,∇ϕm′) → (∇v,∇ϕ),(vm′ · ∇vm′ , ϕm′) → (v · ∇v, ϕ),

(f, ϕm) → (f, ϕ),

and, we obtain in the limit that

ν(∇v,∇ϕ) + (v · ∇v, ϕ) = (f, ϕ).

The existence of an associated pressure p ∈ L , which is uniquely determined in the space L20(Ω) ,

is supplied by Lemma 3.1, below.

ii) Uniqueness: For any solution v ∈ V , there holds

ν‖∇v‖2 = (f, v)− (v · ∇v, v) ≤ ‖f‖−1‖∇v‖,

and, consequently,‖∇v‖ ≤ ν−1‖f‖−1.

Let now v1, v2 ∈ V be two solutions. For the difference w := v1 − v2 , there holds

ν(∇w,∇ϕ) = (v2 · ∇v2 − v1 · ∇v1, ϕ)= ((v2 − v1) · ∇v2, ϕ) − ((v2 − v1) · ∇(v2 − v1), ϕ) + (v1∇(v2 − v1), ϕ)

= (w · ∇v1, ϕ)− (w · ∇w,ϕ) + (v1 · ∇w,ϕ), ϕ ∈ V,

Settingϕ = w , we find

ν‖∇w‖2 = (w · ∇v1, w) ≤ ‖w‖3‖∇v1‖‖w‖6 ≤ c2∗‖∇w‖2ν−1‖f‖−1.

In case that c2∗ν−2‖f‖−1 < 1 this implies that necessarily w = 0 . Q.E.D.

Once a solution v ∈ V of the variational problem (3.1.11) is determined, it remains to showthe existence of a correpondig pressure function p ∈ L , such that

(p,∇ · ϕ) = (f, ϕ) + ν(∇v,∇ϕ) − (v · ∇v, ϕ) ∀ϕ ∈ H. (3.1.14)

The right-hand side of (3.1.14) represents a linear functional l(·) on H with the propertyl(ϕ) = 0, ϕ ∈ V . The existence of a corresponding pressure and its stability is ensured by thefollowing result.

Lemma 3.1 (“inf-sup” inequality): (i) For each linear functional l(·) on H with the prop-erty l(ϕ) = 0, ϕ ∈ J1(Ω) , there exists a uniquely determined p ∈ L , such that

(p,∇ · ϕ) = l(ϕ) ∀ϕ ∈ H, β‖p‖ ≤ supϕ∈H

(p,∇ · ϕ)‖∇ϕ‖ . (3.1.15)

(ii) To each function p ∈ L there exists a function v ∈ H with the property

p = ∇ · v, ‖∇v‖ ≤ β‖p‖, (3.1.16)


with a constant β > 0 independent of p . Further there holds the stability inequality (continuous“inf-sup” inequality):

infq∈L

supϕ∈H

(q,∇ · ϕ)‖q‖ ‖∇ϕ‖ ≥ β > 0 . (3.1.17)

Proof: We use a functional analytic argument following Girault/Raviart [27] (Chapter 1); analternative potential theoretical proof can be found in Galdi [7].(i) We embed the present situation into an abstract functional analytic framework. Startingpoint are the Hilbert spaces L2(Ω) and H = H1

0 (Ω)d with the corresponding norms ‖ · ‖

and ‖∇ · ‖ and their dual spaces L2(Ω)∗ ∼= L2(Ω) and H∗ = H−1(Ω)d (consistimg of linearcontinuous functionals). By

〈−grad p, ϕ〉 := (p,divϕ), ϕ ∈ H,

the gradient (in distributional sense) is defined as linear operator −grad : L2(Ω) → H∗ . Thecorresponding adjoint operator is the divergence operator div : H → L2(Ω)∗ ∼= L2(Ω) . Forthe image spaces (range) R(·) and null spaces N(·) of these operators there holds by generalprincioles:

R(grad) = N(div)0 = J1(Ω)0, R(div) = N(grad)0 = J0(Ω).

Here, N(div)0 := χ ∈ H∗ : 〈χ,ϕ〉 = 0, ϕ ∈ N(div) . A deep result from distribution theory(theorem of De Rham) implies that R(grad) ⊂ H∗ is closed. Hence, R(grad) = J1(Ω)

0 , whatimplies the first assertion (i).

(ii) Further there holds N(grad) = span1 , such that the reduced operator

grad : L = L20(Ω) → J1(Ω)

0

is an isomorphism. Since L and J1(Ω)0 are Hilbert spaces, another general result implies that

the operator grad is also an isomorphism, i. e., the inverse operator grad−1

: J1(Ω)0 → L exists

and is bounded:

β‖grad−1(v)‖ ≤ sup

ϕ∈H

〈v, ϕ〉‖∇ϕ‖ .

This shows that for arbitrary p ∈ L there holds

β‖p‖ ≤ supϕ∈H

〈grad(p), ϕ〉‖∇ϕ‖ = sup

ϕ∈H

(p,∇ · ϕ)‖∇ϕ‖ .

With grad also its adjoint operator div : (J1(Ω)0)∗ → L∗ is an isomorphism. The space

J1(Ω)0 ⊂ H∗ can be identified with the orthogonal complement of J1(Ω) in H :

J1(Ω)0 ∼= J1(Ω)

⊥ := v ∈ H : (∇v,∇ϕ) = 0 ∀ϕ ∈ J1(Ω) .

Consequently, for each p ∈ L there is a v ∈ J1(Ω)⊥ such that

p = ∇ · v , β‖∇v‖ ≤ ‖p‖.

This completes the proof. Q.E.D.

Remark 3.1: For illustrating the important surjectiviy of the divergence operator, we give a


heuristic argument. Let p ∈ L = L20(Ω) be give. The Neumann boundary value problem of the

Laplacian,∆z = p in Ω, ∂nv|∂Ω = 0 ,

possesses a unique (weak) solution z ∈ H1(Ω) ∩L20(Ω), which (on smoothly bounded or convex

polygonal domains) possesses H2-regularity and for which the following a priori estimate holds:

‖∇2z‖ ≤ β−1‖p‖.

The function v := ∇z has then obviously the following properties:

∇ · v = p, n · v|∂Ω = 0, β‖∇v‖ ≤ ‖p‖.

Hence the assertion would be proven if additionally τ · v|∂Ω = 0 with all tangential vectors τat ∂Ω .

Remark 3.2: :The case of inhomogeneous boundary data vin 6≡ 0 can be reduced to thesituation considered in Theorem 3.1 by the following standard argument. Consider the flow fieldu := v − vin ∈ V which satisfies the equation

ν(∇u,∇ϕ) + (u · ∇u, ϕ) = ν(∇v,∇ϕ) − ν(∇vin,∇ϕ) + ((v − vin) · ∇(v − vin), ϕ)

= ν(∇v,∇ϕ) + (v · ∇v, ϕ)− (vin · ∇v, ϕ)− (v · ∇vin, ϕ) + (vin · ∇vin, ϕ)= (f, ϕ)− (vin · ∇v, ϕ)− (v · ∇vin, ϕ) + (vin · ∇vin, ϕ)= (f, ϕ)− (vin · ∇u, ϕ)− (vin · ∇vin, ϕ)− (u · ∇vin, ϕ)

and, consequently,

ν(∇u,∇ϕ) + (u · ∇u, ϕ) + (vin · ∇u, ϕ) + (u · ∇vin, ϕ) = (f, ϕ)− (vin · ∇vin, ϕ) ∀ϕ ∈ V.

In order to carry the argument from the proof of Theorem 3.1 over to this situation, we need toderive a bound for any (existing) solution u ∈ V and the V -ellipticity of the bilinear form

a(u, ϕ) := ν(∇u,∇ϕ) + (u · ∇u, ϕ) + (vin · ∇u, ϕ) + (u · ∇vin, ϕ)

for any fixed u ∈ V . For both purposes, we take ϕ = u to obtain in the latter case

a(u, u) = ν‖∇u‖2 + (u · ∇vin, u) = ν‖∇u‖2 − (u · ∇u, vin)≥ ν‖∇u‖2 − ‖u‖6‖∇u‖‖vin‖3 ≥

(ν − c2∗‖vin‖3

)‖∇u‖2.

Hence, we have the desired V -ellipticity, if the global representation of the boundary data isconstructed such that

‖vin‖3 <ν

c2∗.

The remaining argument is left as an exercise.

Remark 3.3: It is remarkable that the (stationary) Navier-Stokes problem, despite its non-linearity, is solvable for all values of the viscosity parameter ν > 0 . These solutions do notneed to be unique in general, only for sufficiently small data. Actually there are many flowconfigurations for which there are multiple solutions for the same set of data. An example is the


flow in the gap between two concentric spheres (so-called “Taylor problem”). For rotating innersphere and fixed outer sphere there appear different flow patterns. For slow rotational speed(“small-data” case) there is a unique solution. Under increasing the rotational speed this baseflow turns unstable and other more complex stationary flow patterns occur (so-called “Taylorroles”), which exist simultaneously to the base flow for the same set of (stationary) data. Ifthe rotational speed is further increased beyond some “critical” value these stationary statesbreak down and the flow turns nonstationary (oscillating “Taylor roles”) which persist for stillstationary data.

The weak solution v, p ∈ (H + vin)×L of the stationary Navier-Stokes problem obtainedby Theorem 3.1 and Lemma 3.1 possesses additional regularity depending on the data of theproblem. On smoothly bounded or convex polygonal (polyhedral) domains, we have v ∈ H2(Ω)d

and p ∈ H1(Ω) , and there holds the a priori extimate

‖∇2v‖+ ‖∇p‖ ≤ cs ‖f‖+ ‖∇2vin‖ . (3.1.18)

The constant cs depends linearly on the viscosity paramter, cs ∼ 1/ν . For the non-trivial proofof this result, we refer to the relevant literature.

Remark 3.4: Finally, we consider the case of general boundary conditions with Γout 6= ∅ andask for the existence of solutions of the corresponding variational formulation (3.1.4) in thefunction space H = H1

0 (Ω,ΓD) . The proof of the central existance Theorem 3.1 essentiallyused the identity (3.1.12), which now only holds in the form

(u · ∇v, v) = 12(n · u, |v|2)Γout .

The boundary integral on the right does not need to be positive, e. g., in the case of dominatinginflow through Γout. On the basis of this relation the proof of Theorem 3.1 only works forsufficiently small data. Whether this is just a weakness of the argument of proof or really aninherent feature of the Navier-Stokes problem is not known. Particularly it is not known whetherthe trivial solution v ≡ 0 and p ≡ 0 are the only solutions of the homogeneous problem

−ν∆v + v · ∇v +∇p = 0, ∇ · v = 0 in Ω,

with the boundary conditions

v|Γin∪Γrigid= 0, ν∂nv − pn|Γout

= 0.

In the case of pure Dirichlet boundary conditions this is clearly the case. Since the “do-nothing”boundary condition is of purely technical nature without solid physical basis and is supposedto model a Poiseuille-like flow pattern (parallel pipe flow), the possible existence of spurioussecondary solutions would be very disturbing and would render the use of this artificial outflowboundary condition questionable. A satisfactory answer to this question is still open.


3.1.3 Iterative solution schemes

For solving the variational Navier-Stokes problem (3.1.4), at first, we consider the followingfunctional iteration

ν(∇vt,∇ϕ) + (vt−1 · ∇vt, ϕ) = (f, ϕ) ∀ϕ ∈ V := J1, (3.1.19)

with a starting value v0 ∈ V . This reduces the nonlinear Navier-Stokes problem to a sequenceof linear Oseen-type problems.

Theorem 3.2: Let the data of the Navier-Stokes problem satisfy the same smallness assumptionq := c2∗ν

−2‖f‖−1 < 1 as in Theorem 3.1, which guarantees the existence of a unique solution.Then, for any starting value v0 ∈ V the functional iteration (3.1.19) generates a sequence(vt)t∈N ⊂ V , which converges to this solution with linear rate:

‖∇(v − vt)‖ ≤ qt

1− q‖∇(v − v0)‖, t ≥ 1. (3.1.20)

Proof: The functional iteration (3.1.19) is well defined, since in virtue of the theorem of Lax-Milgram for vt−1 ∈ V the next iterate vt ∈ V exists. The iteration can be viewed as a fixedpoint iteration vt = G(vt−1) . For the fixed point mapping G(·) : V → V , one derives theestimate

‖∇(G(v) −G(w))‖ ≤ c2∗‖f‖−1

ν2‖∇(v − w)‖, v, w ∈ V.

Hence under the assumption of the theorem the mapping G(·) is a contraction on V and bythe Banach fixed point theorem the corresponding fixed point iteration converges to the uniquefixed point of G(·) , which is the solution of (3.1.4). The details of this argument are posed asan exercise. Q.E.D.

Remark 3.5: Even simpler than the above functional iteration is the fully explicit treatmentof the nonlinearity leading to the iteration scheme

ν(∇vt,∇ϕ) = (f, ϕ)− (vt−1 · ∇vt−1, ϕ) ∀ϕ ∈ V, (3.1.21)

for an suitable starting value v0 ∈ V . In each step of this iteration only Stokes problems haveto be solved. However, the convergence of this simple iteration requires strong restrictions onthe quality of the starting value v0 , which makes this scheme only feasible in the case of verylarge viscosity when the nonlinear term can be viewed as a small perturbation of the leadingStokes term.

Next, we consider the Newton method in function space for solving problem (3.1.4). Thetangent form of the semi-linear form governing the Navier-Stokes problem

a(v;ϕ) := ν(∇v,∇ϕ) + (v · ∇v, ϕ)

is given bya′(v;ψ,ϕ) := ν(∇ψ,∇ϕ) + (v · ∇ψ,ϕ) + (ψ · ∇v, ϕ).


Starting from some v0 ∈ V the Newton iteration formally reads as follows:

a′(vt−1; vt, ϕ) = a′(vt−1; vt−1, ϕ) + (f, ϕ)− a(vt−1;ϕ) ∀ϕ ∈ V. (3.1.22)

In concrete terms this iteration reads:

ν(∇vt,∇ϕ) + (vt−1 · ∇vt, ϕ) + (vt · ∇vt−1, ϕ)

= ν(∇vt−1,∇ϕ) + (vt−1 · ∇vt−1, ϕ) + (vt−1 · ∇vt−1, ϕ)

+ (f, ϕ)− ν(∇vt−1,∇ϕ)− (vt−1 · ∇vt−1, ϕ)

= (vt−1 · ∇vt−1, ϕ) + (f, ϕ).

In contrast to the functional iteration (3.1.19) here the solvability of the linear subproblems ineach iteration step is not so clear, as the governing bilinear form

a′(vt−1;ψ,ϕ) = ν(∇ψ,∇ϕ) + (vt−1 · ∇ψ,ϕ) + (ψ · ∇vt−1, ϕ)

is generally not V -elliptic,

a′(vt−1;ϕ,ϕ) = ν(∇ϕ,∇ϕ) + (vt−1 · ∇ϕ,ϕ) + (ϕ · ∇vt−1, ϕ)

= ν‖∇ϕ‖2 + ((∇vt−1)Tϕ,ϕ).

The matrix (∇vt−1)T is not positive definite since its trace tr =∑d

i=1 ∂ivt−1i = ∇ · vt−1 = 0 ,

i. e., there are positive as well as negative eigenvalues. Therefore, the existence of the Newtonsequence (vt)t∈N generally requires some smallness assumption on the data of the problem.However, the linear subproblems in the Newton steps may be solvable even if the governingbilinear forms a′(vt−1; ·, ·) are not V -elliptic but regular in a more general sense.

Theorem 3.3: Let the data of the Navier-Stokes problem satisfy the same smallness assumptionq := c2∗ν

−2‖f‖−1 < 1 as in Theorem 3.1, which guarantees the existence of a unique solution.Then, for any starting value v0 ∈ V satisfying

‖∇(v − v0)‖ ≤ 1− q

4ν−2c2∗=: ρ < 1,

the Newton iteration (3.1.22) generates a sequence (vt)t∈N , which converges to this solution withquadratic rate:

‖∇(v − vt)‖ ≤ ρ2t

, t ≥ 1. (3.1.23)

Remark 3.6: A sufficiently good starting value v0 for the Newton iteration may be generatedby a finite number of steps of the functional iteration (3.1.19), which under the same conditionson the data converges with linear rate for any starting value v0 ∈ V .

Proof: (i) By assumption, the data of the problem satisfy q := c2∗ν−2‖f‖−1 < 1 , such that

there is a unique solution v ∈ V . This solution admits the estimate

ν‖∇v‖2 = (f, v)− (v · ∇v, v) ≤ ‖f‖−1‖∇v‖


and, consequently, ‖∇v‖ ≤ ν−1‖f‖−1 . The tangent form of a(·; ·) is uniformly Lipschitzcontinuous in the following sense:

|a′(v;ψ,ϕ) − a′(w;ψ,ϕ)| = |ν(∇ψ,∇ϕ) + (v · ∇ψ,ϕ) + (ψ · ∇v, ϕ)− ν(∇ψ,∇ϕ) − (w · ∇ψ,ϕ) − (ψ · ∇w,ϕ)|

= |((v − w) · ∇ψ,ϕ) + (ψ · ∇(v − w), ϕ)|≤ ‖v − w‖3‖∇ψ‖‖ϕ‖6 + ‖ψ‖3‖∇(v −w)‖‖ϕ‖6≤ 2c2∗‖∇(v −w)‖‖∇ψ‖∇ϕ‖.

In virtue of the above assumptions there holds

‖∇v0‖ ≤ ‖∇(v0 − v)‖+ ‖∇v‖ ≤ 1− c2∗ν−2‖f‖−1

4ν−1c2∗+ ν−1‖f‖−1

=1− c2∗ν

−2‖f‖−1 + 4c2∗ν−2‖f‖−1

4ν−1c2∗=

1 + 3c2∗ν−2‖f‖−1

4ν−1c2∗<

ν

c2∗.

Then, for ϕ ∈ V , there holds

a′(v0;ϕ,ϕ) = ν‖∇ϕ‖2 + (ϕ · ∇v0, ϕ)≥ ν‖∇ϕ‖2 − ‖ϕ‖6‖∇v0‖‖ϕ‖3≥

(ν − c2∗‖∇v0‖

)‖∇ϕ‖2 ≥ α‖∇ϕ‖2, α > 0,

i. e., we have V -ellipticity such that the first iterate v1 ∈ V is well defined. Starting from thisinitial result, we prove the assertion by induction.

(ii) Suppose now that for some t ≥ 1 the iterate vt−1 ∈ V exists and satisfies

‖∇(v − vt−1)‖ ≤ ρ :=1− q

4ν−1c2∗=

1− c2∗ν−2‖f‖−1

4ν−1c2∗.

Then, as in the case t = 1 , we conclude that

‖∇vt−1‖ ≤ ν

c2∗,

and from this the V -ellipticity of a′(vt−1; ·, ·) , which ensures that the next iterate vt ∈ V iswell defined. We have to show that ‖∇(v − vt)‖ ≤ ρ . Then, by induction the whole sequence(vt)t≥0 of Newton iterates exists in V .


(iii) Using the equations determining the solution v and the Newton iterate vt , we find:

a′(v; v − vt, ϕ) = a′(v; v − vt, ϕ)− a′(vt−1; v, ϕ) + a′(vt−1; v, ϕ)

= a′(v; v − vt, ϕ)− a′(vt−1; v − vt, ϕ)− a′(vt−1; vt, ϕ) + a′(vt−1; v, ϕ)

= a′(v; v − vt, ϕ)− a′(vt−1; v − vt, ϕ)− a′(vt−1; vt−1, ϕ)

− (f, ϕ) + a(vt−1;ϕ) + a′(vt−1; v, ϕ)

= a′(v; v − vt, ϕ)− a′(vt−1; v − vt, ϕ)

− a(v;ϕ) + a(vt−1;ϕ) + a′(vt−1; v − vt−1, ϕ)

= a′(v; v − vt, ϕ)− a′(vt−1; v − vt, ϕ)

−∫ 1

0

a′(vt−1 + s(v − vt−1); v − vt−1, ϕ) − a′(vt−1; v − vt−1, ϕ)

ds,

Consequently, by the above Lipschitz continuity estimate,

|a′(v; v − vt, ϕ)| ≤ 2c2∗(‖∇(v − vt−1)‖‖∇(v − vt)‖+ ‖∇(v − vt−1)‖2

)‖∇ϕ‖.

Now, taking ϕ = v − vt in the relation

a′(v; v − vt, ϕ) = ν(∇(v − vt),∇ϕ) + (v · ∇(v − vt), ϕ) + ((v − vt) · ∇v, ϕ)

gives us

a′(v; v − vt, v − vt) ≥ ν‖∇(v − vt)‖2 − |((v − vt) · ∇v, v − vt)|≥ ν‖∇(v − vt)‖2 − ‖v − vt‖3‖∇v‖‖v − vt‖6≥ ν‖∇(v − vt)‖2 − c2∗‖∇v‖‖∇(v − vt)‖2

≥(ν − c2∗‖∇v‖

)‖∇(v − vt)‖2.

Combining the foregoing estimates and observing ‖∇v‖ ≤ ν−1‖f‖−1 and c2∗ν−2‖f‖−1 < 1 , we

conclude

‖∇(v − vt)‖ ≤ 2ν−1c2∗1− c2∗ν

−2‖f‖−1

(‖∇(v − vt−1)‖‖∇(v − vt)‖+ ‖∇(v − vt−1)‖2

)

To simplify this relation, we set at := ‖∇(v − vt)‖ and use the constant ρ−1 = 4ν−1c2∗/(1 −c2∗ν

−2‖f‖−1) from above to obtain

at ≤ 12ρ

−1(atat−1 + a2t−1).

By assumption, there holds at−1 ≤ ρ and therefore,

at ≤ 12ρ

−1(atat−1 + a2t−1) ≤ 12at +

12a

2t−1,

and, consequently, at ≤ a2t−1 ≤ ρ2 < ρ. Finally iterating this inequality, we obtain that

at ≤ ρ2t

, t ≥ 1,



3.2 Finite element discretization

In this section, we present standard finite element methods for solving the stationary Navier-Stokes equations. In contrast to the existence theory developed above the finite element dis-cretization bases on the coupled variational formulation involving velocity v and pressure p asunknowns. The reason is the difficulty in constructing fully conforming finite element subspacesVh ⊂ V . Let Hh ⊂ H and Lh ⊂ L be finite element subspaces on families of structural reg-ular decompositions Thh>0 , which satisfy the following discrete “inf-sup stability” condition(motivated by its continuous counterpart in Lemma 3.1):

minχh∈Lh

maxϕh∈Hh

(χh,∇ · ϕh)‖χh‖‖∇ϕh‖

≥ βh > 0, (3.2.24)

with possibly h-dependent constants βh > 0 . Then, the approximate problems read as follows:Find vh, ph ∈ Hh × Lh , such that

ν(∇vh,∇ϕh) + n(vh, vh, ϕh)− (ph,∇ · ϕh) = (f, ϕh) ∀ϕh ∈ Hh, (3.2.25)

(χh,∇ · vh) = 0 ∀χh ∈ Lh, (3.2.26)

where the modified nonlinear form b(·, ·, ·)) is defined by

n(vh, vh, ϕh) :=12n(vh, vh, ϕh)− 1

2n(vh, ϕh, vh), n(vh, vh, ϕh) = (vh · ∇vh, ϕh)

This modification of the nonlinearity on the discrete level is used in order to carry the argumentfrom the proof of Theorem 3.1 over to the discrete level, since there holds

n(vh, ϕh, ϕh) = 0, vh, ϕh ∈ Hh.

Otherwise, we would have to make the assumotion that the mesh size h is taken sufficientlysmall. This modification is compatible with the continuous level as there holds (exercise)

n(v, ψ, ϕ) = 12 (v · ∇ψ,ϕ) + 1

2 (v · ∇ϕ,ψ) = (v · ∇ψ,ϕ), v ∈ V, ψ, ϕ ∈ H.

3.2.1 General “Stokes elements”

First, we consider the approximation of the (linear) variational Stokes problem: Find v, p ∈H × L , such that

(∇v,∇ϕ)− (p,∇ · ϕ) = (f, ϕ) ∀ϕ ∈ H, (3.2.27)

(χ,∇ · v) = 0 ∀χ ∈ L. (3.2.28)

with normalized viscosity ν = 1 on a convex polygonal (or polyhedral) domain Ω ⊂ Rd (d =

2, 3) . For simplicity, we assume purely homogeneous Dirichlet boundary conditions, i. e., ∂Ω =Γrigid and therefor H := H1

0 (Ω)d and L := L2

0(Ω) . Under these conditions, the (unique)solution of this problem possesses the additional regularity v, p ∈ H2(Ω)d×H1(Ω) and thereholds the a priori estimate

‖∇2v‖+ ‖∇p‖ ≤ cs‖f‖. (3.2.29)

3.2 Finite element discretization 85

The variational Stokes problem has the structure of a saddle point problem and is also called a“mixed” variational formulation. It can be obtained by the Euler-Lagrange approach from theconstrained minimization problem

min J(v) := 12‖∇v‖2 − (f, v) on V ⊂ H,

where the pressure p plays the role of a Lagrange multiplier.

Let Hh ⊂ H := H10 (Ω)

d and Lh ⊂ L := L20(Ω) be finite element subspaces, which satisfy the

discrete “inf-sup stability” condition (3.2.24). Then, the approximate problems read as follows:Find vh, ph ∈ Hh × Lh , such that

(∇vh,∇ϕh)− (ph,∇ · ϕh) = (f, ϕh) ∀ϕh ∈ Hh, (3.2.30)

(χh,∇ · vh) = 0 ∀χh ∈ Lh. (3.2.31)

In this general context, we do not make any assumption on the shape- or size-uniformity ofthe family of decompositions Thh>0 . The finite element spaces Hh ×Lh are generally called“Stokes elements” and particularly “conforming” because Hh ⊂ H and Lh ⊂ L . Later, we willalso consider so-called “non-conforming” Stokes elements with Hh 6⊂ H but still Lh ⊂ L . Wefurther introduce the spaces

Vh := vh ∈ Hh | (∇ · vh, χh) = 0 ∀χh ∈ Lh,

of discrete “solenoidal” vector fields. Notice that Vh 6⊂ V in general. The discrete Stokesproblems (3.2.27)-(3.2.28) are uniquely solvable. This follows from the fact that any solutionvh, ph of the corresponding homogeneous problem satisfies

‖∇vh‖ = 0, (ph,∇ · ϕh) = 0 ∀ϕh ∈ Hh,

which in view of the stability relation (3.2.24) implies that v, p = 0, 0 . For the convergencevh, ph → v, p is is necessary that the approximating spaces Vh are sufficiently “large” forapproximating the space V . To this end, we require the following “minimal” approximationproperty for the spaces Hh × Lh ⊂ H × L :

minϕh∈Hh

‖∇(v − ϕh)‖+ minχh∈Lh

‖p − χh‖ ≤ ch‖∇2v‖+ ‖∇p‖

≤ ch‖f‖. (3.2.32)

Lemma 3.2 (Special best-approximation property): The conforming approximation (3.2.30)-(3.2.31) of the Stokes problem (3.2.27)-(3.2.28) possesses the “(quasi)-best-approximation prop-erty” with respect to the spaces Vh :

‖∇(v − vh)‖ ≤ c

minϕh∈Vh


‖p − χh‖. (3.2.33)

Proof: Subtracting the discrete Stokes equations from their continuous counterparts, we obtainthe following Galerkin orthogonality relations:

(∇(v − vh),∇ϕh)− (p− ph,∇ · ϕh) = 0, ϕh ∈ Hh. (3.2.34)

(χh,∇ · (v − vh)) = 0, χh ∈ Lh. (3.2.35)


Since vh ∈ Vh it follows that for arbitrary ϕh ∈ Vh and χh ∈ Lh there holds

‖∇(v − vh)‖2 = (∇(v − vh),∇(v − ϕh) + (∇(v − vh),∇(ϕh − vh))

= (∇(v − vh),∇(v − ϕh) + (p − ph,∇ · (ϕh − vh))

= (∇(v − vh),∇(v − ϕh) + (p − χh,∇ · (ϕh − v)) + (p− χh,∇ · (v − vh)).

Using the Young inequality ab ≤ a2 + 14b

2 , we conclude that

‖∇(v − vh)‖2 ≤ c‖∇(v − ϕh‖2 + ‖p− χh‖2

,

which implies the assertion. Q.E.D.

The estimate (3.2.33) is unsatisfactory since it is not clear how to construct proper approxi-mations ihv ∈ Vh for v ∈ V . This, however, can be accomplished if the pairs Hh×Lh ⊂ H×Lfor h > 0 are uniformly “inf-sup stable”:

minχh∈Lh

maxϕh∈Hh


≥ β∗, h > 0, (3.2.36)

with a fixed constant β∗ > 0 . This property requires the spaces Hh to be sufficiently “large”(higher dimensional) compared to Lh .

Lemma 3.3: If the pairs Hh × Lh ⊂ H × L, h > 0 , satisfy the uniform “inf-sup” condition(3.2.36), then there also holds the “adjoint” inequality

minϕh∈Hh

maxχh∈Lh


≥ β∗. (3.2.37)

Further, for v ∈ V there holds:

minϕh∈Vh

‖∇(v − ϕh)‖ ≤ (1 + cβ−1∗ ) min

ϕh∈Hh

‖∇(v − ϕh)‖. (3.2.38)

Proof: (i) First, we prove (3.2.37). The relation

(Bhϕh)(χh) := (χh,∇ · ϕh) ∀χh ∈ Lh,

defines a linear operator Bh : Hh → L∗h . The corresponding adjoint operator B∗

h : Lh → H∗h is

defined by(B∗

hχh)(ϕh) := (Bhϕh)(χh) = (χh,∇ · ϕh) ∀ϕh ∈ H.

Since the spaces Hh and Lh are finite dimensional, we have the following relations:

R(Bh) = N(B∗h)

⊥, R(Bh)⊥ = N(B∗

h),

R(B∗h) = N(Bh)

⊥, R(B∗h)

⊥ = N(Bh),

where R(·) = range(·) and N(·) = kernel(·) . The stability inequality (3.2.36) implies that

‖B∗hχh‖H∗ = sup

ϕh∈Hh

(B∗hχh)(ϕh)

‖∇ϕh‖= sup

ϕh∈Hh

(χh,∇ · ϕh)‖∇ϕh‖

≥ β∗‖χh‖, χh ∈ Lh.

Hence the operator B∗h is injective, i. e., R(Bh)

⊥ = N(B∗h) = 0 with bounded inverse B∗−1

h :


N(Bh)⊥ ⊂ H∗ → Lh :

‖B∗−1h ‖ ≤ β−1

∗ .

Then, the adjoint Bh : H → N(B∗h)

⊥ ⊂ L∗h is also invertible with the same bound,

‖B−1h ‖ ≤ β−1

∗ .

This in turn implies

β∗‖∇ϕh‖ ≤ ‖Bhϕh‖L∗

h= sup

χh∈Lh

(Bhϕh)(χh)

‖χh‖= sup

χh∈Lh

(χh,∇ · ϕh)‖χh‖

,

what was to be shown.

(ii) It remains to show (3.2.38). Let v ∈ V , vh ∈ Vh, and ϕh ∈ Hh arbitrary. Then, in virtueof (3.2.37) there holds

‖∇(v − vh)‖ ≤ ‖∇(v − ϕh)‖+ ‖∇(ϕh − vh)‖

≤ ‖∇(v − ϕh)‖+ β−1∗ sup

χh∈Lh

(χh,∇ · (ϕh − vh))

‖χh‖

= ‖∇(v − ϕh)‖+ β−1∗ sup

χh∈Lh

(χh,∇ · (ϕh − v))

‖χh‖≤ (1 + cβ−1

∗ )‖∇(v − ϕh)‖.

Because of the arbitrary choice of vh ∈ Vh and ϕh ∈ Hh this implies (3.2.38). Q.E.D.

Theorem 3.4 (General best approximation property): The conforming approximation (3.2.30)-(3.2.31) of the Stokes problem (3.2.27)-(3.2.28) possesses in case of uniform “inf-sup” stability(3.2.36) the following “(quasi)-best approximation” property:

‖∇(v − vh)‖+ ‖p − ph‖ ≤ c

minϕh∈Hh


‖p− χh‖, (3.2.39)

‖v − vh‖ ≤ ch

minϕh∈Hh


‖p − χh‖. (3.2.40)

Together with the approximation property (3.2.32) this implies the following error estimates:

‖∇(v − vh)‖+ ‖p− ph‖ ≤ ch‖f‖, (3.2.41)

‖v − vh‖ ≤ ch2‖f‖. (3.2.42)

Proof: Subtracting the discrete Stokes equations and their continuous counterpart, we againobtain the Galerkin orthogonality relations

(∇(v − vh),∇ϕh)− (p− ph,∇ · ϕh) = 0, ϕh ∈ Hh. (3.2.43)

(χh,∇ · (v − vh)) = 0, χh ∈ Lh. (3.2.44)

(i) We begin with the estimate of ‖∇(v − vh)‖ . For this, we will not use the result of Lemma3.3. We rather prepare for the argument used below in the context of so-called “stabilized”


Stokes elements. For arbitrary ϕh ∈ Hh we obtain with help of (3.2.43) that


= (∇(v − vh),∇(v − ϕh) + (p− ph,∇ · (ϕh − vh))

= (∇(v − vh),∇(v − ϕh) + (p− ph,∇ · (ϕh − v)) + (p− ph,∇ · (v − vh)),

and further with help of (3.2.44) with arbitrary χh ∈ Lh

‖∇(v − vh)‖2 = (∇(v − vh),∇(v − ϕh) + (p− ph,∇ · (ϕh − v)) + (p − χh,∇ · (v − vh)).

Using Young’s inequality ab ≤ ε2a2 + (4ε2)−1b2 , we conclude

‖∇(v − vh)‖ ≤ c(1 + ε−1)‖∇(v − ϕh)‖+ ‖p − χh‖

+ ε‖p − ph‖. (3.2.45)

(ii) Next, we estimate ‖p − ph‖ . With help of the “inf-sup stability” (3.2.24), we get

‖p − ph‖ ≤ ‖p− χh‖+ ‖χh − ph‖

≤ ‖p− χh‖+ β−1∗ sup

ψh∈Hh

(χh − ph,∇ · ψh)‖∇ψh‖

≤ ‖p− χh‖+ β−1∗ sup

ψh∈Hh

(χh − p,∇ · ψh)‖∇ψh‖

+ β−1∗ sup

ψh∈Hh

(p − ph,∇ · ψh)‖∇ψh‖

≤ c(1 + β−1∗ )‖p − χh‖+ β−1

∗ supψh∈Hh

(p− ph,∇ · ψh)‖∇ψh‖

.

Further, the Galerkin orthogonality relation (3.2.44) implies

‖p− ph‖ ≤ c(1 + β−1∗ )‖p − χh‖+ β−1

∗ supψh∈Hh

(∇(v − vh),∇ψh)‖∇ψh‖

≤ c(1 + β−1∗ )‖p − χh‖+ c‖∇(v − vh)‖.

Combining this estimate with (3.2.45) gives us

‖∇(v − vh)‖ ≤ c‖∇(v − ϕh)‖+ ‖p − χh‖

+ εc(1 + β−1

∗ )‖p− χh‖+ c‖∇(v − vh)‖.

By sufficiently small choice of ε > 0 ,

‖∇(v − vh)‖ ≤ c‖∇(v − ϕh)‖+ (1 + β−1

∗ )‖p − χh‖.

Since ϕh ∈ Hh and χh ∈ Lh are arbitrary, we obtain the first estimate (3.2.39).

(iii) For estimating ‖v − vh‖ , we again use a duality argument. Let z, q ∈ H × L be thesolution of the auxiliary Stokes proble

(∇ϕ,∇z)− (q,∇ · ϕ) = (ϕ, v − vh)‖v − vh‖−1 ∀ϕ ∈ H, (3.2.46)

(χ,∇ · z) = 0 ∀χ ∈ L. (3.2.47)


That is in H2(Ω)d ×H1(Ω) and satisfies the a priori estimate

‖∇2z‖+ ‖∇q‖ ≤ c‖v − vh‖‖v − vh‖−1 = c. (3.2.48)

Setting ϕ := v − vh and using Galerkin orthogonality with the natural nodal interpolationihz ∈ Hh and jhq ∈ Lh , we find

‖v − vh‖ = (∇(v − vh),∇z)− (q,∇ · (v − vh))

= (∇(v − vh),∇(z − ihz)) + (∇(v − vh),∇ihz)− (q − jhq,∇ · (v − vh))

= (∇(v − vh),∇(z − ihz)) + (p − ph,∇ · ihz)− (q − jhq,∇ · (v − vh))

= (∇(v − vh),∇(z − ihz)) + (p − ph,∇ · (ihz − z))− (q − jhq,∇ · (v − vh)).

Further, we estimate

‖v − vh‖ ≤ ‖∇(v − vh)‖‖∇(z − ihz)‖ + ‖p− ph‖‖∇ · (ihz − z)‖+ ‖q − jhq‖‖∇ · (v − vh)‖≤ ch‖∇(v − vh)‖‖z‖H2 + ch‖p − ph‖‖z‖H2 + ch‖q‖H1‖∇(v − vh)‖≤ ch

‖∇(v − vh)‖+ ‖p − ph‖

.


(I) Examples of “conforming” Stokes elements

In the following the various Stokes elements are described by specifying its nodal values forvelocity and pressure ansatz. We restrict the presentation to the 2d case. In most cases there arenatural analogues in 3D. We use the notation PH(T ) and PL(T ) for the local polynomial ansatzfor velocity (H) and pressure (L), respectively, and the superscripts “c” and “dc” for indicatingwhether the global ansatz is “continuous (c)” or “discontinuous (dc)”. In the following the meshfamilies Thh>0 are assumed to be “shape uniform”. We will use the following technical resulton local H1-stable interpolation in 2D and 3D.

Lemma 3.4: There exist generalized interpolation operators I(1)h : H → H

(1)h into the space

H(1)h ⊂ H of conforming cellwise linear or bi/tri-linear finite elements on triangular/tedrahedral

or quadrilateral/hexahedral meshes, such that for v ∈ H there holds

‖v − I(1)h v‖T + hT ‖∇I(1)h v‖T ≤ chT ‖v‖H1(T ), T ∈ Th, (3.2.49)

where T := ∪T ′ ∈ Th |T ′ ∩ T 6= ∅ .

Proof: The proof uses cell or edge averages of v in the interpolation rather than (not H1-stable)point values. For the very technical details, we refer to the literature (e. g., Girault/Raviart [27]or Brenner/Scott [34]). Q.E.D.

a) Stokes elements with discontinuous pressure:

(i) The triangular P c1/P

dc0 element (a) and the quadrilateral Qc

1/Pdc0 element (b):

a) PH(T ) := P1(T )2. PL(T ) := P0(T ) ;

b) PH(T ) := Q1(T )2, PL(T ) := P0(T ) ;


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...............................................................................................................................................................................................................................................................................................................................................................................................................................................

v

p

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.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

v

p

Figure 3.1: The conforming Stokes elements of type P c1/P

dc0 (left) and Qc

1/Pdc0 (right).

The simplest P c1 /P

dc0 element is not useful, as here the subspace Vh of “discrete solenoidal”

velocities may be trivial. The Qc1/P

dc0 element is very popular in engineering because of its

simplicity but turns out to be unstable in general, which may result in unphysical oscillationsin the pressure approximations, so-calle “checkerboard” modes (exercise).

(ii) The triangular P c2/P

dc0 element (a), the extended triangular P c

2 /Pdc1 element (b), and the

quadrilateral Qc2/P

dc1 element (c):

a) PH(T ) := P2(T )2, PL(T ) := P0(T ) ;

b) PH(T ) := P2(T )2 := P2(T )

2 ⊕ spanb1T , b2T , PL(T ) := P1(T ) ;

c) PH(T ) := Q2(T )2, PL(T ) := P1(T ) .

Here b1T = (bT , 0)T and b2T = (0, bT )

T with the (unique) cubic ”bulb function“

bT ∈ P3(T ) : bT |∂K = 0, |T |−1

∫

TbT dx = 1.

Such a “bulb function” can be obtained by the following explicit construction: Let the threesides of a cell T be given by the linear equations li(x) = aix1 + bix2 + ci = 0, i = 1, 2, 3 .Then the cubic polynomial bT (x) := γT l1(x)l2(x)l3(x) with γT = |T |(

∫T l1l2l3 dx)

−1 has all therequired properties.

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.............................................................................................................................................................................................................................................................................................................................................................................................................................................

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...............................................................................................................................................................................................................................................................................................................................................................................................................................................

v

v

v, p

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.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

v

v

× ×

×p

Figure 3.2: The conforming Stokes elements of type P c2/Pdc0 (left), P c2/P

dc1 (middle) and

Qc2/Pdc1 (right).

These Stokes elements satisfy the uniform “inf-sup” stability condition. Actually the proof ofthe stability motivates the use of the extended velocity ansatz P2 . This construction can be


generalized for higher polynomial degrees r ≥ 3 . In these discretizations mass conservation isrealized at least locally on each cell,

∫

∂Tn · vh do =

∫

T∇ · vh dx = 0, T ∈ Th. (3.2.50)

Lemma 3.5 (“inf-sup” stability): The conforming Stokes- elements of type P c2/P

dc0 , P c

2/Pdc1

and Qc2/P

dc1 are uniformly“inf-sup” stable.

Proof: i) The proof bases on the continuous “inf-sup” stability estimate (3.1.17) and the localproperties of the finite element functions used. The single steps of the argument can give hintsfor the construction of stable Stokes elements Let qh ∈ Lh be arbitrarily given. In virtue of thecontinuous stability estimate there holds

β‖qh‖ ≤ supϕ∈H

(qh,∇ · ϕ)‖∇ϕ‖ . (3.2.51)

This implies the relation


(qh,∇ · ϕh)‖∇ϕh‖

‖∇ϕh‖‖∇ϕ‖

+ supϕ∈H

(qh,∇ · (ϕ− ϕh))

‖∇ϕ‖ , (3.2.52)

for arbitrary ϕh ∈ Hh . The way to the asserted discrete stability estimate is then the construc-tion of an interpolation operator πh : H → Hh with the following properties:

(a) (χh,∇ · πhv) = (χh,∇ · v) ∀χh ∈ Lh, (3.2.53)

(b) ‖∇πhv‖ ≤ c1‖∇v‖. (3.2.54)

In view of (3.2.52) this implies


(qh,∇ · πhϕ)‖∇πhϕ‖

‖∇πhϕ‖‖∇ϕ‖

≤ c1 sup

ϕh∈Hh

(qh,∇ · ϕh)‖∇ϕh‖

,

Hence, the discrete “inf-sup” stability estimate is satisfied with β∗ := β/c1 .

(ii) The construction of the operator πh : H → Hh is oriented by the critical properties (a) and(b). By integration by parts there holds

(qh,∇ · v) =∑

T∈|Th

(qh,∇ · v)T =∑

T∈Th

(qh, n · v)∂T − (∇qh, v)T

, qh ∈ Lh.

A local construction of rh has to satisfy the following conditions:

(qh, n · v)Γ = (qh, n · πhv)Γ, Γ ∈ ∂Th, (3.2.55)

(∇qh, v)T = (∇qh, πhv)T , T ∈ Th. (3.2.56)

Here and below, we use the notation ∂Th and ∂2Th for the set of edges and the set of vertices,respectively, of the mesh Th . In each concrete case the above conditions are to be supplementedby additional ones to generate an appropriate operator πh . This involves also the proof of thestability property (b). We will sketch this argument for the simple Stokes elements considered.


(iii) The P c2 /P

dc0 element: The corresponding operator πh is defined by the following local

conditions (observing ∇qh|T ≡ 0 ):

πhv(a) = I(1)h v(a), a ∈ ∂2Th,

(χ, πhv)Γ = (χ, v)Γ, χ ∈ P0(Γ)2, Γ ∈ ∂Th.

The set of 6 nodal functionals (per component) used in this definition is unisolvent with P2(T ) .By construction, there holds

(qh, n · v)Γ = (qh, n · πhv)Γ, Γ ∈ ∂Th.

For establishing the stability of πh , we split πhv|T into its cellwise linear part I(1)h v|T ∈ P1(T )

and a cellwise quadratic part Q(2)h v|T ∈ P2(T ) , such that πhv = I

(1)h v +Q

(2)h v , where

Q(2)h v(a) = 0, a ∈ ∂2Th,

∫

ΓQ

(2)h v ds =

∫

Γ(v − I

(1)h v) ds, Γ ∈ ∂Th.

Splitting Q(2)h v into the contributions from the remaining basis functions corresponding to the

edges Γ and employing the local trace inequality on the cells T , we deduce that (exercise)

‖∇Q(2)h v‖T ≤ c‖∇(v − I

(1)h v)‖T .

From this, we conclude the desired bound

‖∇πhv‖ ≤ ‖∇I(1)h v‖+ ‖∇Q(2)h v‖ ≤ ‖∇I(1)h v‖+ c‖∇(v − I

(1)h v)‖ ≤ c‖∇v‖.

(iv) For the proof of the uniform “inf-sup” stability of the P c2 /P

dc1 element and the Qc

2/Pdc1

element, we refer to the literature (Girault/Raviart [27]). Q.E.D.

Remark 3.7: The existence of a (linear) operator πh : H → Hh with the properties (3.2.53)-(3.2.54) is also necessary for the uniform discrete “inf-sup” stability. To see this, let the stabilitybe given. Then, by Lemma 3.3 there also holds

infvh∈Hh

supχh∈Lh

(χh,∇ · vh)‖∇vh‖‖χh‖

≥ β∗.

For any v ∈ H this implies the existence of a unique element πhv ∈ V Th , such that

(χh,∇ · πhv) = (χh,∇ · v) ∀χh ∈ Lh,

and

β∗‖∇πhv‖ ≤ supχh∈Lh

(χh,∇ · πhv)‖χh‖

≤ supχh∈Lh

(χh,∇ · v)‖χh‖

≤ c‖∇v‖.

This obviously defines as linear operator πh : H → Hh with the desired properties. We empha-size that the explicit construction of such an operator πh is generally more complicated thanin the above examples.

b) Stokes elements with continuous pressure:

These elements are of relatively low dimension but do not possess the local mass conservation


property.

(i) The “P c1 /P

c1 (MINI) element” (a), the “P c

2 /Pc1 (Taylor-Hood) element” (b), and the “Qc

2/Qc1

(isoparametric) element” (c):

a) PH(T ) := P1(T ) := P1(T )2 ⊕ spanb1T , b2T , PL(T ) := P1(T ) ;

b) PH(T ) := P2(T )2, PL := P1(T ) ;

c) PH(T ) := Q2(T )2, PL := Q1(T ) .

Here, again b1T = (bT , 0)T and b2T = (0, bT )

T with the cubic “bulb functions” bT .

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...............................................................................................................................................................................................................................................................................................................................................................................................................................................

v, p

v

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...............................................................................................................................................................................................................................................................................................................................................................................................................................................

v

v, p

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.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

v, p

vv

Figure 3.3: The conforming Stokes elements of type P1⊕ spanb1K , b2K/P c1 (left), P c2/Pc1 (mid-

dle) and Qc2/Qc1 (right).

All three Stokes elements are uniformly “inf-sup” stable. In general, the continuous P cr /P

cr−2

ansatz (for r ≥ 3 ) is “inf-sup” stable. The pairs P cr /Pcr−1 (generalized “Taylor-Hood” elements)

is not stable and requires a local enhancement of the velocity ansatz.

Lemma 3.6 (“inf-sup” stability): The “MINI element” of type P c1/P

c1 as well as the “Taylor-

Hood” elements of type P c2/P

c1 and Qc

2/Qc1 are uniformly “inf-sup” stable.

Proof: The proof uses similar arguments as that of Lemma 3.5. Starting point is again therelation

β‖qh‖ ≤ supv∈H

(qh,∇ · ϕh)

‖∇vh‖‖∇vh‖‖∇v‖

+ supv∈H

(qh,∇ · (v − vh))

‖∇v‖ , (3.2.57)

for arbitrary qh ∈ Lh and vh ∈ Hh . The goal is again the construction of an interpolationoperator πh : H → Hh with the properties (a) and (b). For the MINI element this can beaccomplished similarly as above. Because of the continuity of the pressure ansatz, we have

(qh,∇ · v) = −(∇qh, v).

Hence, for the operator πh has, besides the H -stability, only the property

(∇qh, v) = (∇qh, πhv)T = 0, T ∈ Th,

to be realized. For that, the MINI element provides the two additional bulb components percell in the velocity ansatz. Accordingly, we define the operator πh : H → Hh by the conditions

πhv(a) = I(1)h v(a), a ∈ ∂2Th, (πhv, 1)T = (v, 1)T , T ∈ Th.


The stability estimate is then obtained by using the splitting πhv|T = I(1)h v|T + γT bT with

γT = |T |−1(v − I(1)h v, 1)T (to fulfill the mean value condition):

‖∇πhv‖T ≤ ‖∇I(1)h v‖T + |T |−1|(v − I(1)h v, 1)T |‖∇bT ‖T

≤ ‖∇I(1)h v‖T + |T |−1/2‖v − I(1)h v‖T ‖∇bT ‖T

≤ ‖∇I(1)h v‖T + ch−1T ‖v − I

(1)h v‖T

≤ c‖v‖H1(T ),

which yields‖∇πhv‖ ≤ c‖v‖H1(Ω) ≤ c‖∇v‖.

For the trianglar and quadrilateral Taylor-Hood elements the proof of stability requires a moresophisticated argument, for which we refer to the relevant literature. Q.E.D.

(II) Examples of “nonconforming” Stokes elements

We have seen that the stability condition (3.2.36) does not allow the use of the most naturallowest-order ansatz spaces P c1/P

dc0 and Qc

1/Pdc0 . To enlarge the velocity space, one may turn to

the corresponding “nonconforming”, i. e., not fully continuous, finite elements for the velocity.The elements for the pressure is kept discontinuous in order to preserve local mass conservation.In the case of discontinuous velocity elements all spatial derivatives have to be defines cellwise:

(∇hvh)|T := ∇(vh|T ), T ∈ Th.

With this notation then a pair vh, ph ∈ Hh × Lh is to be determined, such that

(∇hvh,∇hϕh)− (ph,∇h · ϕh) = (f, ϕh) ∀ϕh ∈ Hh, (3.2.58)

(∇h · vh, χh) = 0 ∀χh ∈ Lh. (3.2.59)

For guaranteeing the existence of solutions of these finite dimensional problems and there sta-bility as h→ 0 , we use a discrete analogue of the Poincare inequality,

‖vh‖ ≤ γ‖∇hvh‖, vh ∈ Hh. (3.2.60)

This may be proven for the nonconforming Stokes elements considered by the same argumentas used in the proof of Theorem 3.5, below, and is posed as an exercise. Observing that, for anysolution vh ∈ Hh , there holds

‖∇hvh‖2 − (ph,∇h · vh) = (f, vh),

(ph,∇h · vh) = 0,

we obtain

‖∇hvh‖ ≤ ‖f‖‖vh‖‖∇hvh‖−1 ≤ γ‖f‖. (3.2.61)

This stability result, together with the “inf-sup” stability estimate for the pressure to be proven,particularly implies uniqueness and by that also existence of solutions.

Examples: The nonconforming P nc1 /P dc

0 element (a) and the Qrot1 /P dc

0 elements (b):


a) PH(T ) := P nc1 (T )2, PL(T ) := P0(T ) ;

b) PH(T ) := Qnc1 (T )2, PL := P0(T ) .

with the “rotated bilinear” ansatz on the square reference cell T

Q1(T ) := span1, x1, x2, x21 − x22.

This modification of the usual bilinear ansatz span1, x1, x2, x1x2 is necessary, since the latteris not “unisolvent” with respect to the edge-oriented nodal values (edge midpoint value or edgemean value). This is seen by considering the polynomial p(x) := x1x2 , which vanishes at the4 edge midpoints of the reference cell T = (−1, 1)2 . The modified ansatz Q1(T ) results fromQ1(T ) by a coordinate rotation of 90o (motivating the name “rotated bilinear element”) andis therefore unisolvent. This Stokes element has a natural analogue in 3D:

Q1(T ) := span1, x1, x2, x3, x21 − x22, x22 − x23.

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...............................................................................................................................................................................................................................................................................................................................................................................................................................................

v

p

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.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

vp

Figure 3.4: The nonconforming Stokes elementss of type P nc1 /P0 (left) and Qnc

1 /P0 (right).

For proving the solvability of system (3.2.58)- (3.2.59) in the nonconforming case, we againconsider a solution vh, p ∈ Hh × Lh of the corresponding homogeneous problem, i. e., forf ≡ 0 . For that there holds

‖∇hvh‖ = 0,

which implies ∇vh|T ≡ 0 on each cell T ∈ Th and consequently, in view of the continuityproperties of vh ∈ Hh and the boundary condition, necessarily vh ≡ 0 .

Lemma 3.7 (“inf-sup” stability): The nonconforming Stokes elements of type P nc1 /P dc

0 andQnc

1 /Pdc0 satisfy the uniform “inf-sup” stability condition:

infqh∈Lh

(sup

ϕh∈Hh

(qh,∇h · ϕh)‖qh‖‖∇hϕh‖

)≥ β∗. (3.2.62)

Proof: We construct an interpolation operator πh : H → Hh with the following properties:

‖∇hπhv‖ ≤ c1‖∇v‖, v ∈ H, (3.2.63)

(qh,∇h · πhv) = (qh,∇ · v), v ∈ H, qh ∈ Lh. (3.2.64)


With this construction, we can argue as follows:

β‖qh|| ≤ supϕ∈H

(qh,∇ · ϕ)‖∇ϕ‖ = sup

ϕ∈H

(qh,∇h · πhϕ)‖∇hπhϕh‖

‖∇hπhϕh‖‖∇ϕ‖

≤ c1 sup

ϕh∈Hh

(qh,∇h · ϕh)‖∇ϕh‖

,

what implies the asserted stability estimate (3.2.62) with the constant β∗ := βc−11 .

(i) We begin with the P nc1 /P dc

0 element. The reqirement

∫

Γπhv ds =

∫

Γv ds, Γ ∈ ∂Th,

defines on each cell T ∈ Th a local interpolation operator πTh : H1(T )2 → P1(T )2 . For v ∈ H

the pieces πTh v can be composed to a global function πhv ∈ Hh . The (even cellwise) H1-stability of this interpolation operator can be concluded by observing that ∆πTh v|K ≡ 0 and

∂nπTh v|Γ ≡ const. directly from the construction as follows:

‖∇πTh v‖2T =

∫

∂T∂nπ

Th vπ

Th v ds− (∆πTh v, π

Th v)T =

∫

∂T∂nπ

Th vv ds

= (∇πTh v,∇v)T + (∆πTh v, v)T ≤ ‖∇πTh v‖‖∇v‖T .

(ii) For the Qnc1 /P0 element the argument is similar and left as an exercise. Q.E.D.

For the nonconforming Stokes elements considered the interpolation operator πh is just thenatural nodal interpolation πh = ih : H → Hh . For that, we have the usual error estimate

‖v − ihv‖+ h‖∇h(v − ihv)‖ ≤ cih2‖∇2v‖. (3.2.65)

The following theorem shows that the nonconforming Stokes elements considered allow for errorestimates of optimal order.

Theorem 3.5 (A priori error estimate): For the nonconforming Stokes elements of typeP nc1 /P dc

0 and Qnc1 /P

dc0 (nonparametric) there hold the following a priori error estimates:

‖∇h(v − vh)‖+ ‖p− ph‖ ≤ h‖f‖, (3.2.66)

‖v − vh‖ ≤ ch2‖f‖. (3.2.67)

Proof: We give the proof only for the simpler P nc1 /P dc

0 element. The proof for the Qnc1 /P

dc0

element is posed as an exercise.

(i) Let again eh := v − vh and ηh := p− ph . With an arbitrary ϕh ∈ Hh there holds

‖∇heh‖2 = (∇heh,∇h(v − ϕh)) + (∇heh,∇h(ϕh − vh)),

and for an arbitrary ψh ∈ Hh :

(∇heh,∇hψh) = (∇v,∇hψh)− (∇vh,∇ψh)= (∇v,∇hψh)− (f, ψh)− (ph,∇h · ψh)= (∇v,∇hψh)− (p,∇h · ψh)− (f, ψh) + (ηh,∇h · ψh).


Further, with an arbitrary χh ∈ Lh :

(ηh,∇h · (ϕh − vh)) = (ηh,∇h · (ϕh − v)) + (ηh,∇h · eh)= (ηh,∇h · (ϕh − v)) + (ηh,∇ · v)− (ηh,∇h · vh)= (ηh,∇h · (ϕh − v)) + (p− χh,∇h · eh).

Combining the foregoing relations, we obtain

‖∇heh‖2 ≤ ‖∇heh‖‖∇h(v − ϕh)‖+∆h(v, p)‖∇h(ϕh − v)‖+ ‖∇heh‖

+ ‖ηh‖‖∇(v − ϕh)‖+ ‖p− χh‖‖∇heh‖,

with the “nonconformity term”

∆h(v, p) := maxψh∈Hh

(∇v,∇hψh)− (p,∇h · ψh)− (f, ψh)

‖∇hψh‖.

Using Young’s inequality ab ≤ 14εa

2 + ε−1b2 , for a, b ∈ R+ and arbitrary ε > 0 , we conclude

‖∇heh‖2 ≤ 34ε1‖∇heh‖2 + ε−1

1

‖∇h(v − ϕh)‖2 + ‖p− χh‖2 +∆h(v, p)

2

+ 14ε2‖ηh‖2 +

ε−12 + 1

2

∆h(v, p)

2 + 12‖∇h(v − ϕh)‖2,

and setting ε1 = 1 :

‖∇heh‖2 ≤ 6 minϕh∈Hh

‖∇h(v − ϕh)‖2 + 4 minχh∈Lh

‖p− χh‖2

+ ε−12 + 6∆h(v, p)

2 + ε2‖ηh‖2.(3.2.68)

This intermediate result will be used later on.

(ii) For estimating the pressure error ‖ηh‖ , we use the “inf-sup” stability relation (3.2.62). Withan arbitrary χh ∈ Lh it follows that

‖ηh‖ ≤ ‖p− χh‖+ ‖χh − ph‖

≤ ‖p− χh‖+ β−1∗ max

ϕh∈Hh

(χh − ph,∇h · ϕh)‖∇hϕh‖

≤ ‖p− χh‖+ β−1∗ max

ϕh∈Hh

(χh − p,∇h · ϕh)‖∇hϕh‖

+ β−1∗ max

ϕh∈Hh

(ηh,∇h · ϕh)‖∇hϕh‖

≤ (β−1∗ + 1)‖p − χh‖+ β−1

∗ maxϕh∈Hh


.

Further, there holds

(ηh,∇h · ϕh) = (p,∇h · ϕh)− (ph,∇h · ϕh)= (p,∇h · ϕh)− (∇vh,∇hϕh) + (f, ϕh)

= (p,∇h · ϕh)− (∇v,∇hϕh) + (f, ϕh)− (∇eh,∇hϕh)

and, consequently,

maxϕh∈Hh


≤ ∆h(v, p) + ‖∇heh‖.


This implies

‖ηh‖ ≤ (β−1∗ + 1)‖p − χh‖+ β−1

∗

∆h(v, p) + ‖∇heh‖

. (3.2.69)

Combination of the intermediate results (3.2.68) and (3.2.69) and choice of ε2 :=12β∗ gives us

‖∇heh‖+ ‖ηh‖ ≤ c(β∗)

minϕh∈Hh

‖∇h(v − ϕh)‖+ minχ∈Lh

‖p − χh‖+∆h(v, p), (3.2.70)

with a generic constant c(β∗) ≈ β−1 + 1 > 0 .

(iii) It remains to estimate the nonconformity term ∆h(v, p) . For that, we reformulate observingthat −∆v +∇p = f as follows:

(∇v,∇hψh)− (p,∇h · ψh)− (f, ψh) =∑

T∈Th

(−∆v +∇p− f, ψh)T + (∂nv − pn, ψh)∂T

=∑

T∈Th

(∂nv − pn, ψh)∂T =∑

Γ∈∂Th

(∂nv − pn, ψh)Γ,

where ∂Th again denotes the set of all edges Γ of the cells T ∈ Th . Each side Γ ∈ ∂Th iseither common side of two cells T, T ′ ∈ Th or part of ∂Ω . The function ∂nv − pn on Γ ⊂ Thas in view of the continuity of ∇v and p opposite sign ∂nv − pn on Γ ⊂ T ′ . Consequently,we can write ∑

Γ∈∂Th

(∂nv − pn, ψh)Γ =∑

Γ∈∂Th

(∂nv − pn, [ψh])Γ,

where

[ψh]Γ :=

12 (ψh|Γ∩T − ψh|Γ∩T ′), Γ = T ∩ T ′,

ψh|Γ, Γ ⊂ ∂Ω.

On each side Γ ∈ ∂Th by the special properties of ψh ∈ Hh the jump [ψh] has mean valuezero. Consequently,

(∂nv − pn, [ψh])Γ = (∂nv − pn, [ψh]− [ψh]Γ)Γ = (∂nv − pn− (∂nv − pn)Γ, [ψh]− [ψh]Γ)Γ,

with the mean values

(∂nv − pn)Γ := |Γ|−1

∫

Γ(∂nv − pn) ds, [ψh]Γ := |Γ|−1

∫

Γ[ψh] ds.

Splitting [ψh] into the contributions of the two neighboring cells T, T ′ results in

(∂nv − pn, [ψh])Γ = (∂nv − pn− (∂nv − pn)Γ, ψh|K − ψhΓ)Γ − (∂nv − pn)Γ, ψh|T ′ − ψhΓ)Γ.

Further, by the usual transformation on the reference cell, we see that

‖∂nv − pn− (∂nv − pn)Γ‖Γ ≤ cih1/2‖∇2v‖T + ‖∇p‖, ‖ψh|T − ψhΓ‖Γ ≤ cih

1/2‖∇ψh‖T ,

and analogously for the cell T ′ . From the foregoing results, it now follows that for Γ = T ∩ T ′ :

|(∂nv − pn, [ψh])Γ| ≤ ‖∂nv − pn− (∂nv − pn)Γ‖Γ‖ψh|T − ψhΓ‖Γ + ‖ψh|T ′ − ψhΓ‖Γ

≤ c2h‖∇2v‖T + ‖∇p‖T

‖∇hψh‖T∪T ′ ,


and correspondingly for Γ ⊂ ∂Ω . From this, we conclude

∣∣∣∑

Γ∈∂Th

(∂nv − pn, ψh)Γ

∣∣∣ ≤ ch‖∇2v‖+ ‖∇p‖‖∇hψh‖,

and, consequently, witha constant c > 0 independent of β ,

∆h(v, p) ≤ ch‖∇2v‖+ ‖∇p‖ ≤ ch‖f‖. (3.2.71)

Using the interpolation estimates (proven by the usual transformation argument as in the Lemmaof Bramble/Hilbert)

minϕh∈Hh

‖∇h(v − ϕh)‖+ minχ∈Lh

‖p − χh‖ ≤ ch‖∇2v‖+ ‖∇p‖ ≤ ch‖f‖,

we finally obtain the error estimate (3.2.66).

(iv) For proving the L2-error estimate (3.2.67), we again employ a duality argument. But thedetails are not given here. Q.E.D.

The analysis of the application of nonconforming Stokes elements in the approximationof the (nonlinear) Navier-Stokes problem involves estimation of the modified nonlinear formbh(vh, ψh, ϕh) :=

12(vh ·∇hψ,ϕh)− 1

2(vh ·∇hϕh, ψ) . To this end, besides the “discrete” Poincareinequality, we need discrete versions of some Sobolev inequalities.

Lemma 3.8: For the P nc1 /P dc

0 and the Qnc1 /P

dc0 Stokes elements there holds the following dis-

crete Sobolev inequality in two and three dimensions:

max‖vh‖3, ‖vh‖6 ≤ cnc∗ ‖∇hvh‖, vh ∈ Hh. (3.2.72)

Proof: The proof is posed as exercise. Q.E.D.

3.2.2 Stabilized Stokes elements

There are good reasons to use simple vertex-oriented finite elements in the discretization ofthe Stokes and Navier-Stokes problem in which the degrees of freedom of velocity and pressurelive at the same nodal points. Simplest examples are the conforming P c

1/Pc1 and the Qc

1/Qc1

elements, which however are not “inf-sup” stable by themselves. In the following, we discuss atechnique for stabilizing these elements. We recall the following relation for qh ∈ Lh :


(qh,∇ · ϕh)‖∇ϕh‖

‖∇ϕh‖‖∇ϕ‖

+ supϕ∈H

(qh,∇ · (ϕ− ϕh))

‖∇ϕ‖ , (3.2.73)

where ϕh := i∗hϕ ∈ Hh is an H1-stable (modufied) nodal interpolant (such as I(1)h given by

Lemma 3.4) satisfying on each cell:

‖ϕ − i∗hϕ‖T + hT ‖∇(ϕ− i∗hϕ)‖T ≤ cihT ‖ϕ‖H1(T ),


with the union T of all cells neighboring T . Further, as for the standard nodal interpoalation,there holds for ϕ ∈ H2(Ω) :

‖ϕ − i∗hϕ‖T + hT ‖∇(ϕ− i∗hϕ)‖T ≤ cih2T ‖ϕ‖H2(T ).

For the second term on the right in (3.2.73), we obtain after integration by parts (observeϕ− i∗hϕ ∈ H ):

|(qh,∇ · (ϕ− i∗hϕ))| = |(∇qh, ϕ− i∗hϕ)| ≤∑

T∈Th

‖∇qh‖T ‖ϕ− i∗hϕ‖T

≤ ci∑

K∈Th

hT ‖∇qh‖T ‖ϕ‖H1(T ) ≤ ci

( ∑

T∈Th

h2T ‖∇qh‖2T)1/2( ∑

T∈Th

‖ϕ‖2H1(T )

)1/2

≤ c0ci

( ∑

T∈Th

h2T ‖∇qh‖2T)1/2

‖ϕ‖H1 ≤ c0ci

( ∑

T∈Th

h2T ‖∇qh‖2T)1/2

‖∇ϕ‖,

with a for shape uniform decompositions (Th)h>0 fixed constant c0 ≥ 1 . For the first term onthe right in (3.2.73), we have

supϕ∈H

(qh,∇ · i∗hϕ)‖∇i∗hϕ‖

‖∇i∗hϕ‖‖∇ϕ‖

≤ ci sup

ϕh∈Hh

(qh,∇ · ϕh)‖∇ϕh‖

.

Together the two last estimates imply

βh‖qh‖ ≤ supϕh∈Hh

(qh,∇ · ϕh)‖∇ϕh‖

+( ∑

T∈Th

h2T ‖∇qh‖2T)1/2

, (3.2.74)

with the constant βh := (c0ci)−1β . This result suggests the following modification of the

approximation scheme (3.2.30 - 3.2.31):


(χh,∇ · vh) + sh(χh, ph) = 0 ∀χh ∈ Lh, (3.2.76)

with the “stabilization form”

sh(χh, ph) := α∑

T∈Th

h2T (∇χh,∇ph)T ,

where the constant α > 0 has to be apprpriately chosen.

Lemma 3.9 (Stability): The modified approximation scheme (3.2.75 - 3.2.76) possesses aunique solution vh, ph ∈ Hh × Lh , and there holds the stability estimate

‖∇vh‖+ ‖ph‖+ sh(ph, ph)1/2 ≤ c‖f‖. (3.2.77)

Proof: For proving the existence of solutions it suffices to prove their uniqueness. For that, weset ϕh := vh in (3.2.75) and χh := ph in (3.2.76) to otain

‖∇vh‖2 − (ph,∇ · vh) = (f, vh),

(ph,∇ · vh) + sh(ph, ph) = 0.


Addition of these equations gives us

‖∇vh‖2 + sh(ph, ph) = (f, vh).

In case of homogeneous data, i. e., f ≡ 0 , it follows that vh ≡ 0 and ∇ph ≡ 0 . This alsoimplies ph ≡ 0 for .ph ∈ Lh. Further, since (f, vh) ≤ ‖f‖‖vh‖ ≤ c‖f‖‖∇vh‖ , we conclude thestability estimate

‖∇vh‖+ sh(ph, ph)1/2 ≤ c‖f‖.

In view of the “inf-sup” stability relation (3.2.74), we further obtain

‖ph‖ ≤ c‖f‖,

what completes the proof. Q.E.D.

Theorem 3.6 (Stabilized Stokes elements): For the solution vh, ph ∈ Hh × Lh of themodified approximation scheme (3.2.75 - 3.2.76) there hold the error estimates

‖∇(v − vh)‖+ ‖p− ph‖+ sh(p− ph, p − ph)1/2 ≤ ch‖f‖, (3.2.78)

‖v − vh‖ ≤ ch2‖f‖. (3.2.79)

Proof: Subtracting the discrete Stokes equations from the continuous ones results in

(∇(v − vh),∇ϕh)− (p− ph,∇ · ϕh) = 0, ϕh ∈ Hh. (3.2.80)

(χh,∇ · (v − vh)) + sh(χh, p − ph) = sh(χh, p). (3.2.81)

(i) We begin with the estimation of ‖∇(v − vh)‖ and sh(p− ph, p− ph) . Because of (3.2.80) itfollows, with arbitrary ϕh ∈ Hh :


= (∇(v − vh),∇(v − ϕh) + (p− ph,∇ · (ϕh − vh))

= (∇(v − vh),∇(v − ϕh) + (p− ph,∇ · (ϕh − v)) + (p− ph,∇ · (v − vh)),

and further because of (3.2.81), with arbitrary χh ∈ Lh :

‖∇(v − vh)‖2 = (∇(v − vh),∇(v − ϕh) + (p− ph,∇ · (ϕh − v)) + (p− χh,∇ · (v − vh))

+ (χh − ph,∇ · (v − vh))

= (∇(v − vh),∇(v − ϕh) + (p− ph,∇ · (ϕh − v)) + (p− χh,∇ · (v − vh))

− sh(χh − ph, p− ph) + sh(χh − ph, p)

= (∇(v − vh),∇(v − ϕh) + (p− ph,∇ · (ϕh − v)) + (p− χh,∇ · (v − vh))

− sh(χh − p, p− ph)− sh(p− ph, p − ph) + sh(χh − ph, p).


Therefore, we obtain

‖∇(v − vh)‖2 + sh(p − ph, p − ph) = (∇(v − vh),∇(v − ϕh) + (p − ph,∇ · (ϕh − v))

+ (p− χh,∇ · (v − vh))− sh(χh − p, p− ph) + sh(χh − ph, p)

≤ ‖∇(v − vh)‖‖∇(v − ϕh‖+ ‖p− ph‖‖∇ · (ϕh − v)‖+ ‖p− χh‖‖∇ · (v − vh)‖+ sh(χh − p, χh − p)1/2sh(p− ph, p− ph)

1/2

+ sh(χh − ph, χh − ph)1/2sh(p, p)

1/2

Observing ‖∇p‖ ≤ c‖f‖ it follows that

sh(p, p)1/2 =

(α

∑

T∈Th

h2T ‖∇p‖2T)1/2

≤ ch( ∑

T∈Th

‖∇p‖2T)1/2

= ch‖∇p‖ ≤ ch‖f‖. (3.2.82)

Further, using the local inverse relation for finite elements, we conclude

sh(χh − ph, χh − ph)1/2 =

(α

∑

T∈Th

h2T ‖∇(χh − ph)‖2T)1/2

≤ c( ∑

T∈Th

‖χh − ph‖2T)1/2

≤ c‖p − ph‖+ ‖p− χh‖.

From this, we conclude with help of the inequality ab ≤ ε2a2 + (4ε2)−1b2 :

‖∇(v − vh)‖2 + sh(p− ph, p− ph) ≤ c(1 + ε−1)‖∇(v − ϕh)‖2 + ‖p − χh‖2

+ sh(p− χh, p− χh) + h2‖f‖2+ ε‖p − ph‖2.

(3.2.83)

(ii) Next, we estimate ‖p − ph‖ . By the “inf-sup” stability estimate (3.2.74) and the Galerkinorthogonality relation (3.2.80) it follows

‖p− ph‖ ≤ ‖p − χh‖+ ‖χh − ph‖

≤ ‖p − χh‖+ β−1∗ sup

ψh∈Hh

(χh − ph,∇ · ψh)‖∇ψh‖

+ β−1∗ sh(χh − ph, χh − ph)

1/2

≤ ‖p − χh‖+ β−1 supψh∈Hh

(χh − p,∇ · ψh)‖∇ψh‖

+ β−1 supψh∈Hh

(p− ph,∇ · ψh)‖∇ψh‖


1/2

= ‖p − χh‖+ β−1∗ sup

ψh∈Hh

(χh − p,∇ · ψh)‖∇ψh‖

+ β−1∗ sup

ψh∈Hh

(∇(v − vh),∇ψh)‖∇ψh‖


1/2

and, consequently,

‖p− ph‖ ≤ (1 + β−1∗ )‖p− χh‖+ β−1

∗ ‖∇(v − vh)‖+ β−1∗ sh(χh − ph, χh − ph)

1/2. (3.2.84)

Combining the foregoing estimates with (3.2.83) gives us

‖∇(v − vh)‖2 + sh(p− ph, p− ph) ≤ c(1 + ε−1)‖∇(v − ϕh)‖2 + ‖p− χh‖2

+ sh(p− χh, p− χh) + h2‖f‖2

+ cε(1 + β−1

∗ )2‖p − χh‖2 + β−2∗ ‖∇(v − vh)‖2 + β−2

∗ sh(χh − ph, χh − ph).


Now, fixing ε > 0 sufficiently small, we conclude

‖∇(v− vh)‖+ sh(p− ph, p− ph)1/2 ≤ c

‖∇(v−ϕh)‖+ ‖p−χh‖+ sh(p−χh, p−χh)

1/2 +h‖f‖.

For ϕh = i∗hv and χh = i∗hp , we further obtain

‖∇(v − vh)‖+ sh(p− ph, p − ph)1/2 ≤ ch‖∇2v‖+ ‖∇p‖+ ‖f‖ ≤ ch||f‖.

Therefore, with (3.2.84) it follows that

‖p− ph‖ ≤ ch‖f‖.

(iii) For estimating the L2 error ‖v−vh‖ , we again use a duality argument. Let z, q ∈ H×Lthe solution of the auxiliary “dual Stokes problem”

(∇ϕ,∇z)− (q,∇ · ϕ) = (ϕ, v − vh)‖v − vh‖−1 ∀ϕ ∈ H, (3.2.85)

(χ,∇ · z) = 0 ∀χ ∈ L. (3.2.86)

This is in H2(Ω)d ×H1(Ω) and there holds the a priori estimate

‖z‖H2 + ‖q‖H1 ≤ c‖v − vh‖‖v − vh‖−1 = c. (3.2.87)

With the test function ϕ := v − vh it follows using the Galerkin orthogonality relation withi∗hz ∈ Hh and i∗hq ∈ Lh :

‖v − vh‖ = (∇(v − vh),∇z)− (q,∇ · (v − vh))

= (∇(v − vh),∇(z − i∗hz)) + (∇(v − vh),∇i∗hz)− (q − i∗hq,∇ · (v − vh))

− (i∗hq,∇ · (v − vh))

= (∇(v − vh),∇(z − i∗hz)) + (p− ph,∇ · i∗hz)− (q − i∗hq,∇ · (v − vh))

+ sh(i∗hq, p− ph)− sh(i

∗hq, p)

= (∇(v − vh),∇(z − i∗hz)) + (p− ph,∇ · (i∗hz − z))− (q − i∗hq,∇ · (v − vh))

+ sh(i∗hq, p− ph)− sh(i

∗hq, p).

We further estimate using the estimates for i∗h as follows:

‖v − vh‖ ≤ ‖∇(v − vh)‖‖∇(z − i∗hz)‖+ ‖p− ph‖‖∇ · (i∗hz − z)‖+ ‖q − i∗hq‖‖∇ · (v − vh)‖+ sh(i

∗hq, i

∗hq)

1/2sh(p− ph, p − ph)1/2) + sh(i

∗hq, i

∗hq)

1/2sh(p, p)1/2

≤ ch‖∇(v − vh)‖‖z‖H2 + ch‖p − ph‖‖z‖H2 + ch‖q‖H1‖∇(v − vh)‖+ ch‖∇q‖sh(p − ph, p− ph)

1/2 + ch2‖∇q‖‖∇p‖≤ ch

‖∇(v − vh)‖+ ‖p − ph‖+ sh(p− ph, p− ph)

1/2+ ch2‖f‖.


The error estimates (3.2.78) and (3.2.79) are optimal with respect to the order as well asthe regularity requirements on the solution v, p . The same concept of pressure stabilizationcan also be used for the P c

r /Pcr and Qc

r/Qcr Stokes elements with polynomial degree r ≥ 2 . In

this case the achievable order of approximation is limited by the stabilization term in (3.2.76)to O(h2) . This order barrier can be overcome by choosing the following more consistent stabi-


lization:


(χh,∇ · vh) + sh(χh,∇ph) = gh(χh) ∀χh ∈ Lh, (3.2.89)

with the correction term

gh(χh) := α∑

T∈Th

h2T(∇χh, f)T + (∇χh,∆vh)T

.

In this case the stabilized discrete system is exactly fulfilled by the continuous solution v, p ,such that full Galerkin orthogonality holds.

Remark 3.8: The stabilized P c1/P

c1 Stokes element has a close realtion to the “inf-sup” stable

MINI element. Actuslly there is an algebraic equivalence for appropriately chosen stabilizationparameters (exercise).

Remark 3.9: The stabilization technique described above is a particular case in a whole familyof similar approaches. Though it is universally applicable is suffers from certain shortcomings:

– high computational costs for evaluating the system matrices and possibly additional cou-pling between velocity and pressure unknowns;

– triggering of unphysical flow behavior across “free” outflow boundary.

These defects can partially be lowered by other stabilization techniques, which use edge-orientedstabilisation

sh(χh, ph) = α∑

t∈Th

hT ([∂nχh], [∂nph])∂T ,

where [·] denotes jump across inter-cell boundaries, or by “local projection stabilization” (LPS)

sh(χh, ph) = α∑

T∈Th

(χh − π2hχh, ph − π2hph)T ,

whereπ2h is the projection on a twice coarser mesh.

3.2.3 Navier-Stokes problem: the small-data case

Now, we come back to the finite element approximation of the original Navier-Stokes problem.Again, we restrict the consideration to polygonal/polyhedral domains Ω ⊂ R

d (d = 2, 3) and tothe case of homogeneous Dirichlet boundary conditions v|∂Ω = 0 . At first, we consider againthe small-data case, c2∗ν

−2‖f‖−1 < 1 , in which the continuous problem is guaranteed to possessa unique solution v, p ∈ H ×L . This solution is in H2(Ω)d×H1(Ω) and satisfies the a prioriestimate

‖v‖H2 + ‖p‖H1 ≤ c‖f‖. (3.2.90)

Let Hh × Lh be any of the uniformly “inf-sup” stable Stokes elements introduced above. Inthe following discussion, we concentrate on the “conforming” case, Hh × Lh ⊂ H × L . But


all results obtained also hold true in modified form for the “nonconforming” and “stabilized”Stokes elements. Further, for the reasons discussed above, we use the symmetrized nonlinearform

n(vh, ψh, ϕh) :=12n(vh, ψh, ϕh)− 1

2n(vh, ϕh, ψh), n(vh, ψh, ϕh) := (vh · ∇ψh, ϕh).

With this notation the discrete problems read as follows: Find vh, ph ∈ Hh × Lh , such that

ν(∇vh,∇ϕh) + n(vh, vh, ϕh)− (ph,∇ · ϕh) = (f, ϕh) ∀ϕh ∈ Hh, (3.2.91)

(χh,∇ · vh) = 0 ∀χh ∈ Lh. (3.2.92)

Due to the use of the symmetrized nonlinear form n(·, ·, ·) in the small-data case consideredthe existence of unique solutions can be shown by employing the Brouwer fixed point theoremand the “inf-sup” stability analogously as on the continuous level (exercise). For the followingerror analysis, we assume again that the discrete spaces Hh × LL have the following minimalapproximation properties: There exist interpolation operators ih : H ∩ H2(Ω)d → Hh andjh : L ∩H1(Ω) → Lh , such that

‖w − ihw‖+ h‖∇(w − ihw)‖ ≤ cih2‖w‖H2 , w ∈ H ∩H2(Ω)d, (3.2.93)

‖q − jhq‖ ≤ cih‖q‖H1 , q ∈ L ∩H1(Ω). (3.2.94)

Theorem 3.7 (Error estimates): Under the above assumptions (small data and “inf-sup”stable approximation), there hold the following error estimates:

‖∇(v − vh)‖+ ‖p − ph‖ ≤ c(ν, ‖f‖)h, (3.2.95)

‖v − vh‖ ≤ c(ν.‖f‖)h2. (3.2.96)

Proof: The proof uses similar arguments as that in the context of the linear Stokes problem.Additionally, we have to treat the nonlinearity. For that, we note the following relations forfunctions w,ϕ, ψ ∈ H :

|n(w,ψ, ϕ)| ≤ c2∗‖∇w‖‖∇ϕ‖‖∇ψ‖, (3.2.97)

n(w,ϕ, ϕ) = 0. (3.2.98)

Further, by assumption, we have q := c2∗ν−2‖f‖−1 < 1 . From the equations satisfied by v and

vh , using there the test functions ϕ = v and ϕh = vh , we conclude as usual the estimates

‖∇v‖ ≤ ν−1‖f‖−1, ‖∇vh‖ ≤ ν−1‖f‖−1.

(i) We split the error eh = v− vh into two parts, eh = ξh+ ηh , where ξh := v−wh is the errorin the approximatetion of an auxiliary linearized Stokes problem and ηh := wh − vh representsthe error caused by the presence of the nonlinearity. The auxiliary function wh ∈ Vh is definedas the solution of the Stokes problem

ν(∇wh,∇ϕh) = (f, ϕh)− n(v, v, ϕh) ∀ϕh ∈ Vh. (3.2.99)


Clearly, the corresponding “continuous” solution is v itself. Therefore, by the error estimatesprovided above for the approximation of the Stokes problem, we have the estimate

‖ξh‖+ h‖∇ξh‖ ≤ c(ν, ‖f‖)h2, (3.2.100)

where c(ν, ‖f‖) originates from the constant in the estimate

|(f, ϕ)− n(v, v, ϕ)| ≤ ‖f − v · ∇v‖‖ϕ‖ ≤ c(ν, ‖f‖)‖ϕ‖, ϕ ∈ V,

Next, we estimate ηh . Combining the equations satisfied by wh and vh , we obtain

ν(∇ηh,∇ϕh) = n(vh, vh, ϕh)− n(v, v, ϕh), ϕh ∈ Vh.

and setting ϕh = ηh :

ν‖∇ηh‖2 = n(vh, vh, ηh)− n(v, v, ηh)

= n(vh − v, vh, ηh) + n(v, vh − wh + wh − v, ηh)

= −n(eh, vh, ηh)− n(v, ξh, ηh)

≤ c2∗(‖∇eh‖‖∇vh‖+ ‖∇v‖‖∇ξh‖

)‖∇ηh‖.

This implies‖∇ηh‖ ≤ c2∗ν

−2‖f‖−1‖∇ηh‖+ 2c2∗ν−2‖f‖−1‖∇ξh‖.

By the small-data assumption q := c∗ν−2‖f‖−1 < 1 , we conclude that

‖∇η‖ ≤ 2q

1− q‖∇ξh‖.

Combining this with the estimate for ‖∇ξh‖ derived above yields the desired estimate for‖∇eh‖ .(ii) To estimate the pressure error ‖p−ph‖ , we use the assumed “inf-sup” stability of the spacesHh × Lh . There holds

‖p− ph‖ ≤ ‖p − jhp‖+ ‖jhp− ph‖

≤ ‖p − jhp‖+ β−1∗ sup

ϕh∈Hh

(jhp− ph,∇ · ϕh)‖∇ϕh‖

= ‖p − jhp‖+ β−1∗ sup

ϕh∈Hh

(jhp− p,∇ · ϕh)‖∇ϕh‖

+ β−1∗ sup

ϕh∈Hh

(p− ph,∇ · ϕh)‖∇ϕh‖

≤ (1 + cβ−1∗ )‖p − jhp‖+ β−1

∗ supϕh∈Hh

(p− ph,∇ · ϕh)‖∇ϕh‖

.

Combining the continuous and discrete Navier-Stokes equations, we obtain for ϕh ∈ Hh :

(p− ph,∇ · ϕh) = ν(∇(v − vh),∇ϕh)− n(v, v, ϕh) + n(vh, vh, ϕh)

= ν(∇(v − vh),∇ϕh)− n(v − vh, v, ϕh) + n(vh, vh − v, ϕh),


and further

(p− ph,∇ · ϕh) ≤ ν‖∇(v − vh)‖‖∇ϕh‖+ c2∗(‖∇v‖ + ‖∇vh‖

)‖∇(v − vh)‖‖∇ϕh‖

≤(ν + 2c2∗ν

−1‖f‖−1

)‖∇(v − vh)‖‖∇ϕh‖

= ν(1 + 2q)‖∇(v − vh)‖‖∇ϕh‖.

Combining the foregoing results, we obtain

‖p− ph‖ ≤ (1 + cβ−1∗ )‖p − jhp‖+ ν(1 + 2q)‖∇(v − vh)‖

≤ ch‖p‖H1 + ν(1 + 2q)‖∇(v − vh)‖≤ c(ν, ‖f‖)h.

(iii) Finally, for proving the L2-error estimate, we use again a duality argument. By the small-data assumption the bilinear form (derivative form at the solution v )

a′(v;ψ,ϕ) := ν(∇ψ,∇ϕ) + n(v, ψ, ϕ) + n(ψ, v, ϕ)

is V -elliptic:

a′(v;ϕ,ϕ) = ν‖∇ϕ‖2 + n(v, ϕ, ϕ) + n(ϕ, v, ϕ) ≥ ν‖∇ϕ‖2 − |n(ϕ, v, ϕ)|≥ ν‖∇ϕ‖2 − c2∗‖∇v‖‖∇ϕ‖2 ≥ (ν − c2∗ν

−1‖f‖−1)‖∇ϕ‖2 > 0.

Therefore, the dual problem

ν(∇ϕ,∇z) + n(v, ϕ, z) + n(ϕ, v, z) = (ϕ, eh) ∀ϕ ∈ V.

possesses a unique solution z ∈ V , which is in H2(Ω)2 and satisfies the a priori estimate‖z‖H2 ≤ c‖eh‖ . Setting ϕ = eh yields, with arbitrary zh ∈ Vh :

‖eh‖2 = ν(∇eh,∇z) + n(v, eh, z) + n(eh, v, z)

=[ν(∇eh,∇(z − zh)) + n(v, eh, z − zh) + n(eh, v, z − zh)

]

+[ν(∇eh,∇zh) + n(v, eh, zh) + n(eh, v, zh)

]

=: A+B.

For the first term, we get by the usual arguments

|A| ≤ (ν + 2c2∗ν−1‖f‖−1)‖∇eh‖‖∇(z − zh)‖.

For the second term there holds

B = ν(∇v,∇zh) + n(v, v, zh) + n(v, v, zh)

− ν(∇vh,∇zh)− n(v, vh, zh)− n(vh, v, zh)

= (f, zh) + n(v, v, zh)− (f, zh) + n(vh, vh, zh)− n(v, vh, zh)− n(vh, v, zh)

= n(v − vh, v − vh, zh),

and, consequently,|B| ≤ c2∗‖∇eh‖2

(‖∇(zh − z)‖+ ‖∇z‖

).


Taking now zh = ihz and using the above a priori estimate for ‖z‖H2 , we obtain

‖eh‖2 ≤ c(ν, ‖f‖−1)h‖∇eh‖‖z‖H2 + c‖∇eh‖2‖z‖H2 ≤ c(ν, ‖f‖−1)(h‖∇eh‖+ ‖∇eh‖2

)‖eh‖.

Together with the already proven estimate for ‖∇eh‖ this finally implies the L2-error estimate

‖eh‖ ≤ c(ν, ‖f‖−1)h2,


3.2.4 Transport stabilization for more general data

In the case of “larger” data the Navier-Stokes problem takes on the structure of a diffusion-transport problem with possibly dominant transport. In the usual finite element discretizationthe transport term contributes mainly to the off-diagonals in the system matrix, by which thisloses its definiteness property. On coarser meshes this can result in unphysical oscillations of thediscrete solution and in the failure of iterative solution methods. In order to guarantee stabilityof the discretization, we need extra stabilization of the transport term. This aspect is illustratedfirst in a one-dimensional situation.

3.2.5 A prototypical example in 1D

On the one-dimensional domain Ω = I := (0, 1) ∈ R1 , we consider the linear singularly perturbed

boundary value problem (so-called “Sturm-Liouville Problem”)

−εu′′(x) + q(x)u′(x) = 0, x ∈ I, u(0) = 1, u(1) = 0. (3.2.101)

In the case q ≡ 1 the unique solution has the form (see Fig. 3.5)

uε(x) =e1/ε − ex/ε

e1/ε − 1.

-

6

1

t1

u0(t)uε(t)

ε︸︷︷︸

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Figure 3.5: Solution of the singularly perturbed Sturm-Liouville problem for ε = 0.1 .


In the case ε≪ 1 , we have for x = 1− δ and δ > ε :

uε(1− δ) =e1/ε

e1/ε − 1

(1− e−δ/ε

)≈ 1, sup

x∈I|uε′′(x)| ≈ ε−2,

what justifies the term “boundary layer solution”. For ε→ 0 the limit solution is u0 ≡ 1 , whichdoes not satisfy the boundary condition at x = 1 . The approximation of this problem with theusual centered finite difference scheme (in 1D equivalent to piecewise linear finite elements) withequidistant mesh size h = 1/(N + 1) results in

−(ε+ 12h)yn−1 + 2εyn − (ε− 1

2h)yn+1 = 0, 1 ≤ n ≤ N, y0 = 1, yN+1 = 0.

The corresponding coefficiant matrix is diagonally dominant only under the restrictive condition

h ≤ 2ε. (3.2.102)

For h > 2ε the discrete solution shows an unphysical behavior. To see this, we make the ansatzyn = λn . The possible values for λ are the roots λ± of the quadratic equation

λ2 +2ε

12h− ε

λ+12h+ ε12h− ε

= 0.

Incorporating the boundary conditions y0 = 1 and yN+1 = 0 , we obtain

yn = c+λn+ + c−λ

n−,

and for the coefficients the realtions c++c− = 1 and c+λN+1+ +c−λ

N+1− = 0 , and, consequently,

c− =λN+1+

λN+1+ − λN+1

−

, c+ = 1− λN+1+

λN+1+ − λN+1

−

= − λN+1−

λN+1+ − λN+1

−

.

Hence, the solution looks like

yn =λN+1+ λn− − λN+1

− λn+

λN+1+ − λN+1

−

, n = 0, . . . , N + 1. (3.2.103)

In the present case these roots are given by

λ+,− =−ε±

√ε2 + (12h+ ε)(12h− ε)

12h− ε

=ε∓ 1

2h

ε− 12h, λ+ = 1, λ− =

ε+ 12h

ε− 12h.

For ε≪ 12h . we have λ− ≈ −1 and there results an oscillatory solution:

yn =λn− − λN+1

−

1− λN+1−

, n = 0, . . . , N + 1,

which does not show the qualitatively correct form of the continuous solution. To suppress thisdefect different approaches are available.


(i) Upwind Discretization: The first-order term u′(x) in the differential equation is dis-cretized by one of the following one-sided difference quotients

∆+h u(x) =

u(x+ h)− u(x)

h, ∆−

h u(x) =u(x)− u(x− h)

h.

The choice of the backward difference quotient ∆−h observes the physical transport from the left

to the right (see the form of the limit solution u0(x) ). This results in the difference equation

(−ε+ h)yn−1 + (2ε + h)yn − εyn+1 = 0.

For arbitrary h > 0 the corresponding system matrix is diagonally dominant (even an M -matrix). In this case the ansatz ynλ

n leads to the quadratic equation

λ2 − 2ε+ h

ελ+

ε+ h

ε= 0,

with the roots

λ+,− =2ε+ h

2ε±

√(2ε+ h)2 − 4ε(ε + h) =

2ε+ h± h

2ε, λ+ =

ε+ h

ε, λ− = 1.

The critical root λ+ is now always positive, such that in the discrete solution (3.2.103),

yn =λN+1+ − λn+

λN+1+ − 1

,

no oscillations occur. For general transport coefficient q(x) the “upwinding” has to be chosenlocally in accordance to the sign of qn = q(xn) . This way of discretizing the transport termu′(x) by one-sided difference quotients is called “backward differencing” or “upwinding”. Itlimits the total accuracy of the discretization to first order only even in regions outside theboundary layer where the solution is smooth.

(ii) Artificial Diffusion: Maintaining the central difference approximation of the transportterm u′(x) the diffusion coefficient ε is set to a larger value εh := ε+ δh . This results in thedifference equation

−(εh +12h)yn−1 + 2εhyn − (εh − 1

2h)yn+1 = 0, 1 ≤ n ≤ N.

For the corresponding discrete solution, we obtain again by the above ansatz the form

yn =λN+1+ − λn+

λN+1+ − 1

, λ+ =εh +

12h

εh − 12h.

Obviously, in this case λ+ > 0 for ε+δh > 12h , i. e., for the choice δ ≥ 1

2 . By this modification,we obtain again a diagonally dominant (M -matrix) and, consequently, a stable discretization.However, this approach strongly smears out the boundary layer on the interval [1 − εh, 1] andthe total accuracy of approxination is also limited to first order.

Remark 3.10: The two described strategies for dealing with the lacking stability in case ofdominant transport, simple “upwinding” and “artificial diffusion”, are able to cure the prob-lem but at the expense of limiting the approximation accuracy to first order. Other related


approaches of formally higher oder (such as higher-order one-sided finite differences or higher-order artificial diffusion) do not preserve the strong M -matrix property of the system matrix.This can only be achieved by low-order stabilization techniques.

(iii) Streamline Diffusion (SD-FEM): In the context of finite element discretization thereis an approach which can be viewed as consistent, transport-oriented artificial diffusion. Weconsider the model problem

−εu′′(x) + qu′(x) + αu(x) = f(x), x ∈ I, u(0) = u(1) = 0, (3.2.104)

with q ≡ 1 and α ≥ 0 . In the so-called “streamline diffusion method” the variational formula-tion of this boundary value problem,

ε(u′, ϕ′) + (u′ + αu,ϕ) = (f, ϕ), ∀ϕ ∈ H := H10 (I), (3.2.105)

is modified to

ε(u′, ϕ′) + (u′ + αu,ϕ + δϕ′) = (f, ϕ+ δϕ′), ∀ϕ ∈ H, (3.2.106)

with a parameter function δ , which is coupled to the local mesh size h like 0 < δ ∼ h ≤ 1such that αδ ≤ 1 .

Lemma 3.10: The (nonsymmetric) bilinear form

aδ(u, v) := ε(u′, v′) + (u′ + αu, v + δv′), u, v ∈ H,

is “elliptic”,

aδ(v, v) ≥ 12‖v‖2δ , v ∈ H, (3.2.107)

with respect to the “energy norm”

‖v‖δ :=(ε‖v′‖2 + ‖δ1/2v′‖2 + α‖v‖2

)1/2,

Proof: We have

aδ(v, v) = ε‖v′‖2 + (v′ + αv, v + δv′) = ε‖v′‖2 + ‖δ1/2v′‖2 + α‖v‖2 + (v′, v) + α(v, δv′)

≥ ‖v‖2δ − |(v′, v)| − α|(v, δv′)|≥ ‖v‖2δ − 1

2α‖v‖2 − 12‖δ1/2v′‖2 ≥ 1

2‖v‖2δ .

Here, we have used the assumption αδ ≤ 1 and the identity

∫ 1

0v′(x)v(x) dx =

1

2

∫ 1

0(v2)′(x) dx =

1

2

v2(1)− v2(0)

= 0.


We emphasize that the parameter δ is a function of x (cellwise constant with respect to0 = x0 < . . . < xN+1 = 1 ) and, consequently, appears inside ‖δ1/2v′‖ . Analogously, we use thesymbol h = h(x) for a cellwise constant mesh-size function. The corresponding finite element


Galerkin method (with linear elements) in the subspaces Hh ⊂ H = H10 (0, 1) reads

uh ∈ Hh : aδ(uh, ϕh) = lδ(ϕh) ∀ϕh ∈ Hh, (3.2.108)

with the modified functional lδ(ϕh) := (f, ϕh + δϕ′h) .

Theorem 3.8 (Error estimate for SD-FEM): Suppose that ε ≪ hmin , α = 1 , and thataccordingly the stabilization parameter in the streamline diffusion in each subinterval In ischosen as δn ∼ hn . Then, there holds the a priori error estimate

‖u− uh‖δ ≤ c‖h3/2u′′‖, (3.2.109)

with a constant c independent of ε , h and δ .

Proof: We sketch the proof for the case α = 1 and δ = h. Combination of the variationalequations for u and uh yields the following perturbed Galerkin orthogonality relation for theerror e := u− uh :

aδ(e, ϕh) = ε(u′, ϕ′h) + (u′ + u, ϕh + δϕ′

h)− ε(u′h, ϕ′h)− (u′h + uh, ϕh + δϕ′

h)

= (f, ϕh) + (u′ + u, δϕ′h)− (f, ϕh + δϕ′

h)

= (u′ + u− f, δϕ′h)

= ε(u′′, δϕ′h).

(3.2.110)

With help of the ellipticity (3.2.107) and this orthogonality relation, we obtain:

‖e‖2δ ≤ aδ(e, u− ϕh) + ε(u′′, δ(ϕh − uh)′), (3.2.111)

with arbitrary ϕh ∈ Hh . The first term on the right is estimated by

|aδ(e, u− ϕh)| ≤ ε|(e′, (u− ϕh)′)|+ |(e′ + e, u− ϕh)|+ |(e′ + e, δ(u − ϕh)

′)|≤ ε‖e′‖‖(u− ϕh)

′‖+ ‖δ1/2e′‖+ ‖δ1/2e‖‖δ−1/2(u− ϕh)‖+ ‖δ1/2e′‖+ ‖δ1/2e‖‖δ1/2(u− ϕh)

′‖.

For the choice ϕh := ihu it follows with help of the usual local interpolation estimate observingδ = h ≤ 1 and ε ≤ hmin :

|aδ(e, u− ihu)| ≤ cε‖e′‖‖hu′′‖+ c‖δ1/2e′‖+ ‖e‖‖δ−1/2h2u′′‖+ c‖δ1/2e′‖+ ‖e‖‖δ1/2hu′′‖

≤ cε1/2‖e′‖+ ‖δ1/2e′‖+ ‖e‖‖h3/2u′′‖

≤ 14‖e‖2δ + c‖h3/2u′′‖2.

For the second term in (3.2.111), we get with ϕh := ihu by analogous arguments:

ε|(u′′, δ(ihu− uh)′)| ≤ ε‖δ1/2u′′‖

‖δ1/2(ihu− u)′‖+ ‖δ1/2e′‖

≤ c‖h3/2u′′‖‖δ1/2hu′′‖+ ‖δ1/2e′‖

≤ c‖h3/2u′′‖2 + 14‖e‖2δ .

Now, combination of the derived estimates implies the desired result. Q.E.D.


The error estimate (3.2.109) shows that for a smooth solution u (without boundary layer), orif the mesh is fine enough for resolving the boundary layer, the SD-FEM converges in the “energynorm” with order O(h3/2) , which is higher than the order achievable by simple upwinding orartificial diffusion. However, the corresponding (nonsymmetriuc) system matrix is definite butneither diagonally dominant nor an M -matrix.

Transport stabilization for the Navier-Stokes equations

The methods for transport stabilization described above for the 1d case can also be applied inthe context of the multi-dimensional nonlinear Navier-Stokes equations. We discuss here onlythe potentially higher-order SD-FEM and other approaches of this type, which are the mostcommonly used techniques within finite element discretizations.

(i) Streamline Diffusion (SDS): The idea of “streamline diffusion” is to add artificial diffusiononly in transport (streamline) direction. This can be accomplishes in two different but essentiallyequivalent ways:

a) by augmenting the test functions by transport-oriented terms, what leads to a so-called“Petrov-Galerkin method”, or

b) by adding certain “least-squares” terms in the variational formulation of the problem.

We describe a simple variant of this method for the stationary Navier-Stokes problem this timeallowing nonhomogeneous inflow and outflow data v|Γin∪Γout = vin . The discretization may beby any of the conforming, nonconforming or stabilized Stokes elements discussed above. Forthat, we introduce the following notation for the main terms in the variational Navier-Stokesproblem:

ah(vh, ϕh) := ν(∇hvh,∇hϕh),

nh(vh, vh, ϕh) :=12

(vh · ∇hvh, ϕh)− (vh · ∇hϕh, vh)

,

b(ph, ϕh) := (ph,∇h · ϕh),

and for the additional stabilization terms:

sph(ph, χh)) :=∑

T∈Th

δpT (∇ph,∇χh)T

svh(vh, ϕh) :=∑

T∈Th

δvT(vh · ∇vh, vh · ∇ϕh)T + (∇ · vh,∇ · ϕh)T

,

rph(vh, χh) :=∑

T∈Th

δpT (f + [ν∆vh]− vh · ∇vh,∇χh)T ,

rvh(vh, ph, ϕh) :=∑

T∈Th

δvT (f + [ν∆vh +∇ph], vh · ∇ϕh)T ,

where vh is a suitable reference velocity. The stabilization parameters are chosen as

δpT = αpT ν−1h2T , δvT = αvT max

T|vh|−1hT ,

with suitable damping constants αpT , αvT , usually chosen uniformly or adaptively in the range

[0.1, 1] for all mesh cells for balancing the effect of stabilization and approximation. Then, thediscrete problems read as follows:


Find pairs vh, ph ∈ (vinh +Hh)× Lh , such that

ah(vh, ϕh) + nh(vh.vh, ϕh) + svh(vh, ϕh) + bh(ph, ϕh)

= (f, ϕh) + rvh(vh, ph, ϕh) ∀ϕh ∈ Hh,(3.2.112)

bh(χh, vh) + sph(ph, χh) = rph(vh, χh) ∀χh ∈ Lh. (3.2.113)

This discretization is fully consistent with the continuous Navier-Stokes problem since the com-bined stabilization and correction terms vanish for the continuous solution v, p . On the onehand, this ensures exact Galerkin orthogonality for the errors v − vh, p − ph , but, on theother hand, due to the many extra terms, it makes the computation of the system matrices veryexpensive and introduces additional couplings between velocity and pressure unknowns, whatsignificantly increases the cost of the iterative solution of the resulting algebraic systems. Theexistence of solutions vh, ph ∈ (vinh +Hh×Lh of the system (3.2.112) - (3.2.113) follows anal-ogously as in the non-stabilized case due to the definiteness and smallness of the stabilizationterms. To summerize: The above stabilized discretization serves the following purposes:

– The term ∑

T∈Th

δvT (vh · ∇vh, vh · ∇ϕh)T

stabilizes the transport term in the sense of the directional “streamline diffusion”.

– The term ∑

T∈Th

δvT (∇ · vh,∇ · ϕh)T

enhances the incompressibility condition.

– The term ∑

T∈Th

δpT (∇ph,∇χh)

stabilizes the pressure in case of an unstable “equal-order” Stokes element.

The correction terms only make the discretization consistent. A careful but very technical anal-ysis shows that this discretization actually yields an improved error behavior compared to thesimple low-order stabilization by “upwinding” or “artificial diffusion”.

(ii) Local Projection Stabilization (LPS): An alternative approach to transport stabiliza-tion uses the concept of “local projection” as already employed above in the context of pressurestabilization. The method avoids some of the drawbacks of the streamline diffusion technique(unphysical outward bending of streamlines using the “do nothing” condition at free outflowboundary and high extra costs for the evaluation of the stabilization terms particularly in 3D).In the LPS the stabilization terms

sph :=∑

T∈Th

δpT (∇(ph − π2hph),∇(χh − π2hχh))T ,

svh :=∑

T∈Th

δvT (vh · ∇(vh − π2hvh), vh · ∇(ϕ− π2hϕ))T ,

are used and there is no need for additional correction terms, i. e., rph = rvh = 0 . Here, π2his a local projection or interpolation into the spaces L2h and H2h on the coarser mesh T2h ,


respectively. The stabilization parameters δpT and δvT are chosen similarly as in the streamlinediffusion method. The resulting discretization is formally of second order accurate, the gener-ation of the corresponding system matrices is relatively cheap, and the consistency deffect atoutstream boundaries is avoided.

3.2.6 Treatment of nonlinearity

For dealing with the nonlinearity in the discretized Navier-Stokes problem there are severalpossible approaches following the concepts already described above on the continuous level.For notational symplicity, we describe these methods only for the case possibly with simplepressure stabilization (without correction terms) but without transport stabilization. In allthese iterative methods one starts from a suitable field v0h and generates a sequence of iteratesvth, pth ∈ Hh×Lh , which under certain conditions converges for t→ ∞ to the discrete solutionvh, ph .(i) Stokes linearization (explicit iteration)In case of dominant diffusion ν ≈ 1 (e. g., for highly viscous liquids, small velocities, or smallflow domains), one may use the simple “Stokes iteration”

ν(∇hvth,∇hϕh)− (pth,∇h · ϕh) = (f, ϕh)− n(vt−1

h , vt−1h , ϕh) ∀ϕh ∈ Hh, (3.2.114)

(χh,∇h · vth) + sph(χh, pth) = 0 ∀χh ∈ Lh, (3.2.115)

in which the nonlinearity is treated fully explicitly. In each iteration step only a discrete sym-metric and positive definite Stokes problem has to be solved. As on the continuous levelthis iteration converges under a small-data assumption ( q := c2∗ν

−2‖f‖−1 < 1/2 for con-forming Stokes elements), if additionally the starting value is already sufficiently accurate,‖∇h(v

0h − vh)‖ ≤ ν(1 − 2q)/(2c2∗) < 1 (exercise). These convergence conditions are rather

severe and usually not met in practice.

(ii) Oseen linearization (semi-implicit iteration)The restrictive smallness condition for the starting value in the “Stokes iteration” can be avoidedby using the semi-implicit “Oseen linearization” leading to the following functional iteration:

ν(∇hvth,∇hϕh) + n(vt−1

h , vth, ϕh)− (pth,∇h · ϕh) = (f, ϕh) ∀ϕh ∈ Hh, (3.2.116)

(χh,∇h · vth) + sph(χh, pth) = 0 ∀χh ∈ Lh. (3.2.117)

In the small-data case ( q = c2∗ν−2‖f‖−1 < 1 for conforming, “inf-sup” stable Stokes elements),

this iteration converges for any starting value v0h with linear rate uniformly for h (exercise). Ineach iteration step a linear but nonsymmetric “Oseen problem” has to be solved. This usuallyrequires additional transport stabilization, e. g., the LPS method with terms of the form

svh(vt−1h , vth, ϕh) :=

∑

T∈Th

δvh(vt−1h · ∇(vth − π2hv

th), v

t−1h · ∇(ϕ− π2hϕ))T

(iii) Newton linearizationThe linearization by applying the Newton method leads to an iteration, which in the small-data case (again q = c2∗ν

−2‖f‖−1 < 1 for conforming Stokes elements) converges quadraticallyuniformly in h provided the starting value is good enough. Suitable starting values may beobtained by the globally convergent functional iteration. However, in practice the Newton


method is observed to converge even in cases of more general “larger” data provided the startingvalue is chosen sufficiently close to the wanted (only locally unique) solution. In setting upthe Newton iteration, one usually only consideres the derivative of the Navier-Stokes form andneglects the likewise nonlinear terms in the transport stabilization, since these contain the factorsδvT and are therefore considered as “small”. The resulting iteration then reads as follows:

ν(∇hvth,∇hϕh) + n(vt−1

h , vth, ϕh) + n(vth, vt−1h , ϕh)− (pth,∇h · ϕh)

= (f, ϕh) + n(vt−1h , vt−1

h , ϕh) ∀ϕh ∈ Hh,(3.2.118)

(χh,∇h · vth) + sph(χh, pth) = 0 ∀χh ∈ Lh. (3.2.119)

In each iteration step, one has to solve a linear but nonsymmetric and, in general, indefiniteOseen-like problem. The indefiniteness of the “reaction term” n(vth, v

t−1h , ϕh) may cause the

usual iterative algebraic methods to fail and can make the practical use of this method a verydifficult task. However, dropping this indefinite term reduces the Newton method to the or-dinary functional iteration, which converges only with linear rate and may not be capable toapproximate solutions, which are only locally unique.

3.2.7 Solution of linear discrete problems

We now discuss the solution of the linear discrete problems occurring within the nonlineariterations described above. To this end, we consider again the discretization by “inf-sup” stableor stabilized (conforming or nonconforming) Stokes elements as presented above. Further, werestrict us again to the case of pure homogeneous Dirichlet boundary conditions, ∂Ω = Γrigid .Let

ψih, i = 1, . . . , NH := dimHh, χih, i = 1, . . . , NL := dimLh,be the usual nodal bases of the velocity space Hh and the pressure space Lh , respectively.Then, the linear problems to be solved take on the form of a block system:

Aξ =[A+N1 +N2 B

−BT C

][x

y

]=

[b+ n0 + n2

0

]=: β, (3.2.120)

for the nodal value vectors ξ = x, y in the representations

vh =

NH∑

j=1

xjψjh, ph =

NL∑

j=1

yjχjh,

with the corresponding system matrices and right hand side vectors:

A :=(ν(∇hψ

ih,∇hψ

jh))NH

i,j, B := −

((χih,∇h · ψjh)

)NL,NH

i,j=1,

N1 :=(n(vt−1

h , ψih, ψjh))NH

i,j, N2 =

(n(ψih, v

t−1h , ψjh)

)NH

i,j,

C :=( ∑

T∈Th

δpT(∇h(χ

ih − π2hχ

ih),∇h(χ

jh − π2hχ

jh))T

)NL

i,j=1

b :=((f, ψih)

)NH

i=1,

n0 := −(n(vt−1

h , vt−1h , ψih)

)NH

i=1, n2 =

(n(vt−1

h , vt−1h , ψih)

)NH

i=1.


This block-system has the typical structure of a saddle point problem, in which the matrix A isstrongly indefinite with positive as well as negative eigenvalues in the symmetric case. Here, theStokes iteration corresponds to N1 = N2 = 0 and n2 = 0 , the Oseen (functional) iteration toN2 = 0 and n1 = n2 = 0 , and the Newton iteration to n1 = 0 . The mean value condition onthe pressure, (ph, 1) = 0 , which ensures uniqueness, may be directly built into the nodal basis ofLh , leading to a global coupling of all pressure unknowns. Alternatively, it can be incorporatedthrough an extra equation in the system (ph, 1) =

∑NL

i=1 yi(χih, 1) = 0 , which however, spoils the

symmetry of the matrix A . Therefore, in the following discussion, we prefer the first option.

3.2.8 Schur complement methods

For simplicity, we use the same notation b for the right hand side vector in all three casesb+ n0 (Stokes iteration), b (Oseen functional iteration) and b+ n2 (Newton iteration). Then,the discretized Navier-Stokes problem reads

(A+N1 +N2)x+By = b, (3.2.121)

−BTx+ Cy = 0. (3.2.122)

The matrix A + N1 + N2 is nonsymmetric but usually regular. Consequently, the componentx can be eliminated from the system as follows:

x = −(A+N1 +N2)−1By + (A+N1 +N2)

−1b,

Σy := (BT (A+N1 +N2)−1B + C)y = BT (A+N1 +N2)

−1b.

The matrix Σ := BT (A+N1 +N2)−1B + C is called “Schur complement” of A+N1 +N2 in

the block matrix A . It leads to the following block triangular decomposition:

A =

[A+N1 +N2 B

−BT C

]=

[A+N1 +N2 0

−BT Σ

][I (A+N1 +N2)

−1B

0 I

]. (3.2.123)

This suggests some iterative procedures, which are based on the fact that very efficient solutionmethods are available for the “inversion” of the main-diagonal block A+N1 +N2 .

a) Uzawa Algorithm:

The “classical” method for iteratively solving the saddle point problems (3.2.120) is the “Uzawaiteration”, which operates on the pressure variables y . Starting from some y0 , satisfying themean value condition, one successively computes for t ≥ 1 :

(A+N1 +N2)xt = b−Byt−1, (3.2.124)

yl = yt−1 + θ(BTxl − Cyt−1), (3.2.125)

where θ > 0 is a appropriately chosen relaxation parameter. In order to cope with possi-ble irregularitied of the mesh Th (cell anisotropies or local mesh refinements) the system is“preconditioned” by the mass matrix of the pressure space.

M =ML =((χih, χ

jh

)NL

i,j=1.


The modified Uzawa iteration then reads as follows:

(A+N1 +N2)xt = b−Byt−1, (3.2.126)

Myl =Myt−1 + θ(BTxt − Cyt−1), (3.2.127)

Each iteration step requires essentially the “inversion” of the matrix A +N1 + N2 and of themass matrix M , what can be achieved by variants of the CG-method for nonsymmetric orindefinite matrices or by a multigrid method. Eliminating the velocity variable xl convertsUzawa algorithm into a fixed point iteration of the form:

yt = yt−1 + θM−1(BT (A+N1 +N2)

−1(b−Byt−1)− Cyt−1)

= (I − θM−1Σ)yt−1 + θM−1(BT (A+N1 +N2)−1b).

Hence, the Uzawa algorithm can be interpreted as a damped Richardson iteration for solvingthe Schur complement equation. For this, we obtain in the simplest case N1 = N2 = 0 (i. e.,solution of Stokes problem) by the Banach fixed point theorem the following result.

Theorem 3.9 (Uzawa Algorithm): Let Σ be symmetric. For sufficiently small damping,θ < λmax(M

−1Σ)−1, the Uzawa algorithm converges to the solution x, y of the saddle pointproblem (3.2.120). With

0 < 1− λmin(M−1Σ)

λmax(M−1Σ)= 1− 1

cond2(M−1Σ)=: q < 1,

there holds the error estimate

|yl − y| ≤ qt|y0 − y|, t ≥ 1. (3.2.128)

Proof: The exact solution y satisfies the fixed point equation

y = (I − θM−1Σ)y + θM−1(BTA−1b).

For the iteration error et := y − yt there holds

et = (I − θM−1Σ)et−1, t ≥ 1.

Consequently, we have convergence for arbitrary starting point y0 if and only if the spectralradius of the iteration matrix satisfies spr(I − θM−1Σ) < 1 . This is guaranteed by the choiceof the relaxation parameter θ < λmax(M

−1Σ)−1. The representation

M−1Σ =M−1/2(M−1/2ΣM−1/2)M1/2

implies by a similarity transformation that the eigenvalues of the matrix M−1Σ equal those ofthe symmetric and positive matrix M−1/2ΣM−1/2 . Consequently, all eigenvalues of M−1Σ arereal and positive and, by the choice of θ there holds

spr(I − θM−1Σ) = max1− θλ(M−1Σ) ≤ 1− λmin(M−1Σ)

λmax(M−1Σ)< 1.

The assertion then follows from the general results for fixed point iterations. Q.E.D.


For the symmetric case N1 = N2 = 0 , we shall see below that cond2(M−1Σ) ≤ 1 uniformly

for all h . Therefore the Uzawa algorithm converges linearly with mesh-independent rate pro-vided that the relaxation parameter θ is chosen sufficiently small. By varible choice of θ = θtthe speed of convergence can be optimized. Then, the Uzawa algoithm corresponds to the or-dinary “gradient method” applied to the Schur complement equation. We do not consider thisin more detail since, next, we shall look at the much faster “conjugate gradient (CG) method”.The Uzawa algorithm may also be applied in the case N1 6= 0 (Oseen functional iteration) asthen the Schur complement M−1Σ is definite (though nonsymmetric). However, in the caseN2 6= 0 (Newton iteration) it may fail to converge.

b) CG-type methods

We consider again the symmetric case Σ = ΣT . As in the Uzawa algorithm the system matrixis preconditioned with the pressure mass matrix ML , i. e., the CG method is used in form of aPCG method for the system

M−1Σy =M−1(BTA−1b). (3.2.129)

This is equivalent to applying the CG method to the symmetric and positive definite matrix(modulo the mean value zero condition) M−1/2ΣM−1/2 . The speed of the CG method isdetermined by the spectral condition number

κ := cond2(M−1Σ) =

λmax(M−1Σ)

λmin(M−1Σ)

like

|yt − y| ≤ κ(1− κ−1/2

1 + κ−1/2

)t|y0 − y|, t ∈ N, (3.2.130)

where y0 is the starting value of the iteration.

Theorem 3.10 (Schur complement): For the Schur complement Σ = BTA−1B there holds

condnat(M−1Σ) ≤ c0

β2h, (3.2.131)

with the constant βh > 0 in the “inf-sup” stability inequality of the Stokes element Hh/Lh usedand c0 ≤ 5 , assuming that in case of a stabilized Stokes element the stabilization parametersδT are chosen such that δT ‖∇qh‖2T ≤ ‖qh‖2T . In the case of conforming, “inf-sup” stable Stokeselements, we have c0 = 1 .

Proof: We note the identities

λmin(M−1Σ) = min

y∈RNL

〈Σy, y〉〈My, y〉 = min

y∈RNL

〈BTA−1By + Cy, y〉〈My, y〉

= miny∈RNL

〈AA−1By,A−1By〉〈My, y〉 +

〈Cy, y〉〈My, y〉

.


For any scalar product on RNH such as 〈·, ·〉A := 〈A·, ·〉 , there holds

|x|A := maxz∈RNH

〈x, z〉A|z|A

, x ∈ RNH .

Using these relations for x := A−1By yields


y∈RNL

|A−1By|2A〈My, y〉 +


= miny∈RNL

maxz∈RNH

〈AA−1By, z〉2〈Az, z〉〈My, y〉 +


= miny∈RNL

maxz∈RNH

〈By, z〉2〈Az, z〉〈My, y〉 +


.

In view of the definition of the matrices A , B , C , and M we find via the assiciation y ∈RNL ↔ ph ∈ Lh and z ∈ R

NH ↔ ψh ∈ Hh that


y∈RNL

maxz∈RNH

(qh,∇h · ψh)2‖qh‖2‖∇hψh‖2

+1

‖qh‖2∑

T∈Th

δT ‖∇qh‖2T=: β2h.

Analogously,

λmax(M−1Σ) = max

y∈RNL

〈Σy, y〉〈My, y〉

= maxqh∈Lh

maxψh∈Hh

(qh,∇h · ψh)2‖qh‖2‖∇hψh‖2

+1

‖qh‖2∑

T∈Th

δT ‖∇qh‖2T≤ 5,

where, we use that in 3D:‖∇h · ψh‖2 ≤ c1‖∇hψh‖2,

with c1 = 4 in the general case. For conforming. “imf-sup” stavle elements, Hh ⊂ H , this canbe sharpened to c1 = 1 . This completes the proof. Q.E.D.

As byproduct of the above proof, we obtain the following norm bound for the Schur comple-ment:

‖M−1Σ‖ ≤ c1, (3.2.132)

where c1 ≤ 5, in general, and c1 = 1 for conforming, “inf-sup” stable Stokes elements.

In the practical realization of the CG methods for the matrix M−1Σ it is to be observedthat each iteration step consists essentially of a matrix-vector multiplication with M−1Σ andthe evaluation of several scalar products. This requires as most “expensive” step the solution ofa linear system with the “Laplace-like matrix” A :

y →M−1Σy ⇔ y → By → A−1By → (BTA−1B + C)y →M−1(BTA−1B + C)y.

Since A−1 is not available exactly the evaluation of A−1By has to be done iteratively. Forthe matrix A (system matrix of the discretization of the vector-Laplace operator) there existvery efficient and robust PCG- or multigrid methods even on very irregular meshes. Usually


these “inner” iterations within the “outer” CG iteration are controlled by an adaptive stoppingcriterion oriented by the corresponding iteration residual. This means that in the outer CGiteration the matrix M−1A−1 used in the defect computation is faulty (e. g., with elementwiseerrors of size O(10−8) ) and changes permanently in the course of the iteration. Therefore,the conditions for the good convergence of the outer CG iteration are not fully satisfied, whatmay result in erratic convergence behavior and residuals remaining above the accuracy levelO(10−8) of the inner iteration. This undesirable defect can be cured by embedding the PCGiteration for the Schur complement M−1Σ into an outer defect correction iteration. Thereby,the “unprecise” PCG iteration is used as “preconditioner” S of a simple, robust Richardsoniteration:

yt → dt := M−1Σyt −M−1(BTA−1b) → rt = S−1dt → yt+1 := yt + rt.

In this way, one obtains a simple, robust, and efficient solution method for the discretized Stokesproblem. The extension of this approach to the nonsymmetric and indefinite cases A+N1 (Oseenfunctional iteration) A+N1 +N2 (Newton iteration) using generalized PCG methods such asthe “GMRES or the “biCGstab method has also not been very successful yet. However, thereare competing multigrid methods, which are directly applied to the full block-system matrixA in the saddle point problem (3.2.120) and are more efficient than the described “stabilized”Schur complement-CG iteration. Here, special care has to be taken in choosing an appropriatesmoothing iteration, which can cope with the indefinite character of the problem.

3.2.9 Multigrid method

The main idea underlying a “multigrid algorithm” consists in the fast elimination of “high-frequency” components of the error on the finest mesh (“smoothing”) by “cheap” relaxationmethods (e. e., point–Jacobi or Gauß-Seidel method) and the reduction of the remaining “smooth”low-frequency error part by defect correction on coarser meshes (“coarse-grid correction”). Webriefly describe this process.

The multigrid iteration uses a hierarchy of finite element subspaces,

H0 × L0 ⊂ . . . ⊂ Hl × Ll ⊂ . . . ⊂ HL × LL,

which is obtained within a systematic (adaptively controlled) mesh refinement process. Theconnection between these spaces is given by so-called “prolongation operators” P ll−1 : Hl−1 ×Ll−1 → Hl×Ll and “restriction operators” Rl−1

l : Hl×Ll → Hl−1×Ll−1 . In the finiteelement context these operators are simply taken as

P ll−1 : natural embedding, Rl−1l : L2 projection.

The main component of a multigrid algorithm is the smoothing operation Sl : Hl × Ll →Hl×Ll on the various mesh levels 0 ≤ l ≤ L (l = 0 corresponding to the coarsest and l = Lto the finest mesh.). The multigrid iteration

Mξ = M(l, z0, ξ) (3.2.133)

on level l with starting value z0 and m1 pre- and m2 post-smoothing steps is recursivelydefined by:


Multigrid Algorithm M(l.z0, ξ) for l ≥ 0 :

For l = 0 the multigrid algorithm consists in the “exact” solution of the coarsest problem, i. e.,M(0, z0, ξ) := A−1ξ . For l ≥ 1 the following iteration is performed:

1. Pre-smoothing m1-times: z1 := Sm1

l z0.

2. Residual on level l : rl := ξ −Alz0.

3. Restriction to level l − 1 : rl−1 := Rl−1l rl.

4. Coarse-grid corretion starting with q0 := 0 : q := M(l − 1, q0, rl−1).

5. Prolongation to level l : z2 := z1 + P ll−1q.

6. Post-smooting m2-times: M(l, z0, ξ) := Sm2

l z2.

Is the multigrid algorithm applied γ-times on each mesh level, one speaks in case γ = 1 of “V -cycle” and in case γ = 2 of “W -cycle”. The variants with γ ≥ 3 are to expensive and not usedin practice. In non-standard situations (e. g., strongly nonsymmetric systems) the V -cycle isusually not robust enough so that the more robust but also more expensive W -cycle is prefered.However, if the multigrid iteration is only used as an “inner” iteration for preconditioning arobust “outer” iteration (e. g., the GMRES method) one usually employs the cheaper V -cycle.The so-called “F -cycle” is an attractive compromise between V - and W -cycle.

4v3v

v2

v1

v0

v4

vv3

2

v1

v0

Figure 3.6: Schemes of multigrid V -cycle (upper left), F -cycle (oper right) andW -cycle (bottom)

It is known that a multigrid algorithm as described above, if applied in the standard fi-nite element discretization of scalar elliptic model problems such as the Poisson problem, is of“(almost) optimal complexity”. This means that on a mesh Th with Nh nodal unknowns thediscrete solution uh is obtained with only O(NhL(Nh)) operations (uniformly in h ). Thiscomplexity is improved to the optimal O(NL) if the starting value for the multigrid iteration

is taken as the solution obtained on the preceding coarser mesh, u(0)L := uL−1 . Unfortunately,

analogous theoretical results are not available yet for saddle point problems such as the Navier-Stokes equations. However, numerical experience shows that even in such indefinte situationsthe multigrid concept can work surprisingly well.


The design of a muligrid algorithm for solving a saddle point problem requires special care.In particular, the choice of the smoothing operation is difficult since the common fixed pointiterations such as the point-Jacobi or point-Gauß-Seidel method do not work in this case. Thisproblem can be handles in different ways.

Block-Gauß-Seidel smoothing (“Vanka smoothing”): A very popular smoothing for the (generallynonsymmetric) block matrix A is obtained by cellwise blocking of velocity and pressure variableswithin a global Gauß-Seidel iteration. The purpose of this trick is to realize at least locally theindefinite coupling of velocity and pressure variables. This approach was originally proposed fora finite difference discretization of the Stokes problem and has proven very successful also in thecontext of the finite element method.

We briefly describe the realization of this idea for the nonconforming “rotated” Qnc1 /P

dc0

Stokes element. The velocity and pressure variables in a cell T or a patch of cells are numberedconsecutively and the corresponding element system matrices indicated by the subscript “loc”.These local degrees of freedom are then simultaneously updated within a blockwise Gauß-Seideliteration:

Slocvkloc +Blocp

kloc = “known”, BT

locvkloc = “known”,

where Sloc := Aloc , Sloc := (A+N1)loc , or Sloc := (A+N1 +N2)loc . This iteration runs overall cells in the current mesh Th . The local Stokes-like problems have the dimension dloc = 9(in 2D) and dloc = 19 (in 3D). The corresponding matrix in 2D is shown below.

×

×

×

× × : nodes for v

© : nodes for p

Aloc =

Sloc,1 O Bloc,1

O Sloc,2 Bloc,2

−BTloc,1 −BT

loc,2 0

.

For minimizing costs the main diagonal blocks Sloc,i may by reduced to diagonal matrices by“lumping”, Sloc,i ≈ Dloc,i . Further, for enhancing robustness the iteration may be damped,vk+1h = vkh + ω(vk+1

h − vkh) with some ω ∈ (0, 1) .

We illustrate the performance of the resulting multigrid algorithm for the “rotated” Qnc1 /P

dc0

Stokes element applied in approximating the so-called “lid-driven cavity” problem by the Oseenlinearization for a range of viscosities 1 ≥ ν ≥ 1/5000 .

# cells 1600 6400 25600 # iterations

ν = 1 0.081 0.096 0.121 4

ν = 1/100 0.098 0.099 0.130 6

ν = 1/1000 0.227 0.245 0.168 9

ν = 1/5000 0.285 0.368 0.370 18

Table 3.1: Multigrid convergence rates (2 pre- and 1 post-smoothing with the “Vanka smoother”)and number of outer functional iteration steps on uniformly refined meshes.


We remark that a similar “blockwise” iteration can also be used in an incomplete block-LUdecomposition. From the common analysis of multigrid methods we know that “pointwise”iterations lose their smoothing property if the mesh has large anisotropies. This is the standardcase in resolving boundary layers. The solution is the use of special smoothing, which in thelimit of extremly streched cells reduce to “exact” solvers. These are, for example, “line-Gauß-Seidel” and “ILU” iteration. Since the above “Vanka smoother” acts on the velocity variableslike a “point-Gauß-Seidel” iteration, there occur problems in case of strongly streched meshcells. This difficulty can be overcome by patchwise blocking the variables in the direction of the“anisotropic” mesh refinement leading to a method called “stringwise Gauß-Seidel smoothing”.

3.3 A nested solution scheme

The practical solution process of the Navier-Stokes problem may be organized in form of a“nested” solution scheme, in which discretization (by a stable Stokes element), linearization (bya Newton-type iteration) and algebraic solution (by a GMRES-multigrid method) is adaptivelycoupled. The whole process should be controlled by a posteriori error estimates for the differentalgorithmic components. We formulate this scheme for a nonlinear problem of the followingabstract variational form:

a(u;ϕ) = f(ϕ) ∀ϕ ∈ V, (3.3.134)

where the nonlinear form a(·; ·) represents the “energy form” of the Navier-Stokes problem,with V = H × L and u = v, p ∈ H × L , or that of a general quasi-linear elliptic problem,with V = H1

0 (Ω) ∩W 1,∞(Ω) .

Let be given a desired tolerance TOL for some error measure E(·) (e. g., a norm ‖ · ‖ orsome more local functional E(u) = u(a) ) and a maximum mesh complexity Nmax (due to thecapacity limits of the computer used). Starting from a coarse initial mesh T0 , a hierarchy ofsuccessively refined meshes Tl, l ≥ 1 , with Nl := #T ∈ Tl , and corresponding finite elementspaces Vl, with Vl ⊂ Vl+1 , are generated by the following algorithm.

Nested (adaptive) solution algorithm:

(0) Initialization for l = 0 : Compute an initial approximation u0 ∈ V0 (coarsest mesh).

(1) Defect correction iteration for l ≥ 1: Start with u(0)l := ul−1 ∈ Vl .

(2) Iteration step: For j ≥ 0 form the defect functional

d(j)l (ϕ) := f(ϕ)− a(u

(j)l ;ϕ), ϕ ∈ Vl. (3.3.135)

Pick a suitable approximation a′(u(j)l ; ·, ·) to the derivative form a′(u

(j)l ; ·, ·) (with good

stability and solvability properties) and compute a correction v(j)l ∈ Vl from the linear

equation

a′(u(j)l ; v

(j)l , ϕ) = d

(j)l (ϕ) ∀ϕ ∈ Vl. (3.3.136)

For that, a CG/GMRES-type method with multigrid preconditioning may be employedusing the hierarchy of already constructed meshes Tl, ...,T0. This “inner” iteration

3.3 A nested solution scheme 125

yields an approximation v(j)l ≈ v

(j)l together with an a posteriori error estimate

E(v(j)l − v

(j)l ) ≤ ηiniter,

from which a stopping criterion is obtained by balancing iteration and discretization errors

on mesh Tl . Then, update u(j+1)l = u

(j)l + λlv

(j)l , with some relaxation parameter λl ∈

(0, 1] , increment j and go back to (2). This process is repeated until an approximation

ul := u(j)l ∈ Vl is reached with a sufficiant accuracy, also based on an a posteriori estimate

for this “outer” iteration,E(ul − ul) ≤ ηoutiter.

(3) Error estimation and mesh adaptation: Accept ul as the solution on mesh Tl and evaluatean a posteriori error estimate of the form

E(ul − u) ≤ ηdiscr(ul) =∑

T∈Tl

ηT (ul).

The so-called “cell-error indicators” ηT (ul) are used to construct a new (refined) meshTl+1 and corresponding finite element space Vl+1 by seeking equilibration through thefollowing strategy:

if ηT ≫ TOL

Nlrefine T, if ηT ≈ TOL

Nlkeep T, if ηT ≪ TOL

Nlcoarsen T.

where “refining” and “coarsening” may be realized using the structure of the hierarchicalmeshes Tl−1 ⊂ Tl ⊂ Tl+1 (bysection allowing “hanging” nodes). Here, the underlyingphilosophy is that the additional “iteration errors” due to the use of an only approximatefinite element solution ul ≈ ul ∈ Vl can be controlled by a combined a posteriori errorestimate of the form

E(ul − u) ≤ ηdiscr + ηoutiter + ηiniter.

This induces stopping criteria for the outer and inner algebraic iterations of the form

ηoutiter + ηiniter ≤ κηdiscr, (3.3.137)

where usually, for safety reasons, κ := 1/10 . If then, on some mesh Tl good equilibrationis achieved, there holds

E(ul − u) ≈∑

T∈Tl

ηT ≈∑

T∈Tl

TOL

Nl= TOL,

and the solution process can be stopped. The process is also stopped if the current meshTl has already maximal complexity, i. e., Nl ≈ Nmax . Otherwise, if ηdiscr > TOL andNl < Nmax , increment l and go back to (1).

The nested solution process described above requires to choose several parameters in thestopping criteria for the inner and outer algebraic iterations and in the strategy for equilibratingthe cell-error indicators on the current mesh. The appropriate choice of these parameters dependsvery much on the problem to be solved and needs some expert knowledge by the user. There


exists a systematic approach to deriving a posteriori error estimates of the form (3.3.137) evenfor nonlinear problems as considered here. The theory underlying the so-called “Dual WeightedResidual (DWR) method will be the subject of the next chapter.

Remark 3.11: Usually the evaluation of the a posteriori error estimate (3.3.137) involves onlythe solution of linearized problems. Hence, the whole error estimation may amount to only arelatively small fraction of the total cost for the solution process. This has to be compared to theusually much higher cost when working on non-adapted meshes and without efficient stoppingcriteria.

3.4 Exercises

Exercise 3.1: Let (vm)m∈N be a bounded sequence of functions in V = J1(Ω) , which convergesweakly in V and strongly in J0(Ω) to some limit v ∈ V :

(∇vm,∇ϕ) → (∇v,∇ϕ), ϕ ∈ V, ‖vm − v‖ → 0 (m → ∞).

For any fixed ϕ ∈ V let (ϕm)m∈N be a sequence, which converges strongly in V to ϕ , i. e.,‖∇(ϕm − ϕ)‖ → 0 (m → ∞) . Show that this implies the following convergences:

(∇vm,∇ϕm) → (∇v,∇ϕ), (vm · ∇vm, ϕm) → (v · ∇v, ϕ) (m→ ∞).

Exercise 3.2: The key property of the nonlinear form b(u, v, w) := (u · ∇v,w) in the Navier-Stokes equations is the relation

b(u, v, v) = 0, u, v ∈ V := v ∈ H10 (Ω)

d |∇ · v = 0.

If in the approximation (non-conforming) finite element spaces Vh 6⊂ V are used this property islost on the discrete level. Therefore, one may use instead a symmetrized version of this nonlinearform, such as

b(u, v, w) := 12b(u, v, w) − 1

2b(u,w, v).

Show that for this modification there holds

b(u, v, w) = b(u, v, w), u, v, w ∈ V,

and, by construction,b(uh, vh, vh) = 0, uh, vh ∈ Vh.

Exercise 3.3: A common linearization of the Navier-Stokes equations is the so-called ‘’Oseenlinearization”, with a given divergence-free flow field v ∈ H1(Ω)d ,

−ν∆v + v · ∇v +∇p = f, ∇ · v = 0, in Ω,

and the usual boundary conditions on ∂Ω = Γrigid ∪ Γin ∪ Γout :

v = 0 on Γrigid, v = v on Γin, −ν∂nv + pn = 0 on Γout.

Show the unique weak solvability of this linear problem for the case that meas(Γrigid) > 0 andv · n ≥ 0 on Γout . Hint: One may use the theorem of Lax-Milgram.

3.4 Exercises 127

Exercise 3.4: It has been shown in class that the variational stationary Navier-Stokes problemin R

d (d = 2, 3) ,

ν(∇v,∇ϕ) + (v · ∇v, ϕ) = (f, ϕ) ∀ϕ ∈ V := v ∈ H10 (Ω)

d |∇ · v = 0,

possesses a unique solution v ∈ V under the smallness condition c2∗ν−2‖f‖−1 < 1 on the data.

Show that under the same condition for any starting value v0 ∈ V the functional iteration

ν(∇vt,∇ϕ) + (vt−1 · ∇vt, ϕ) = (f, ϕ) ∀ϕ ∈ V,

produces a sequence (vt)t∈N ⊂ V , which converges in V to this solution. Hint: One mayinterprete the functional iteration as a fixed point iteration vt = g(vt−1) and employ the Banachfixed point theorem.Remark: This offers an alternative proof for the existence of weak solutions of the Navier-Stokesequations in the case of small data.

Exercise 3.5: Consider the approximation of the variational (linear) Stokes problem in twodimensions,

v ∈ V = J1(Ω) : (∇v,∇ϕ) = (f, ϕ) ∀ϕ ∈ V,

by the finite element method on quasi-uniform families of triangulations.

a) Construct V -conforming finite element subspaces Vh ⊂ V from the quintic H2-conformingArgyris plate element. (Hint: Observe that div(rot) = 0 .)

b) Specify appropriate local nodal bases of these spaces Vh .

c) For the case f ∈ L2(Ω)2 and Ω a convex polygonal domain derive optimal order errorestimates in the H1 and L2 norms.

d) Give an idea on how the corresponding discrete pressures ph can be computed once thevelocity field vh is known.

Exercise 3.6: Show that the Qc1/P

dc0 Stokes element is not “inf-sup” stable in general. To

this end consider a uniform cartesian (quadrilateral) mesh Th of the unit square Ω = (0, 1)2

and show that for the so-called “checkerboard” pressure function qcheckh ∈ Lh , which has thealternating constant values ±1 there holds

(qcheckh ,∇ · ϕh) = 0, ϕh ∈ Hh.

This shows that in this case the “inf-sup” stability estimate cannot hold.

Exercise 3.7: Let the Navier-Stokes problem be approximated by a conforming and uniformly“inf-sup” stable Stokes element with finite element subspaces Hh × Lh ⊂ H × L . Supposethat the data are sufficiently small, c2∗ν

−2‖f‖−1 < 1 , to guarantee the existence of a uniquesolution v, p . For solving the corresponding discrete nonlinear problems consider the followingiteration, in which the nonlinear term is treated fully “explicitly”,

ν(∇vth,∇ϕ)− (pth,∇ · ϕ) = (f, ϕ)− 12(v

t−1h · ∇vt−1

h , ϕ) + 12(v

t−1h · ∇ϕ, vt−1

h ) ∀ϕ ∈ Hh,

(χ,∇ · vth) = 0 ∀χ ∈ Lh,


for a starting value v0h ∈ Vh , which reduces the solution of a nonlinear algebraic system to asequence of linear algebraic (Stokes) systems.

a) Give a conditions on the size of the initial error ‖vh − v0h‖ , which implies that this iterationconverges and specify the rate of convergence.

b) Define a modified iteration, which under the same condition on the data converges for allstarting value v0h ∈ Vh (with proof).

Exercise 3.8: Let Thh>0 be a shape uniform family of triangulations in R2 and v ∈ H . In

class it has been claimed that for the (cellwise) quadratic polynomial Q(2)h v|T ∈ P2(T ) with the

properties

Q(2)h v(a) = 0, a ∈ ∂2Th,

∫

ΓQ

(2)h v ds =

∫

Γ(v − I

(1)h v) ds, Γ ∈ ∂Th,

there holds‖∇Q(2)

h v‖ ≤ c‖∇(v − I(1)h v)‖.

Give a proof of this estimate. (Hint: Consider each cell T ∈ Th separately, write Q(2)h v in its

nodal basis representation and use the trace inequality locally on T .)

Exercise 3.9: Consider the special situation of a family of uniformly cartesian (quadrilateral)meshes in 2D and the discretization of the Stokes problem by the nonconforming “rotated” bi-linear Stokes element Qnc

1 /Pdc0 .

a) Show that on any cell T the prescription of the nodal values χΓ(v) := |Γ|−1(vh, 1)Γ, Γ ⊂ ∂T ,uniquely determines a polynomial in the space Qrot

1 (T ) = span1, x1, x2, x21 − x22 (“unisol-vence”).

b) Use the technics described in class for proving the uniform “inf-sup” stability of this Stokeselement.

Exercise 3.10: It has been shown in class that on triangular meshes in 2D the conformingP c1 /P

c1 (MINI) Stokes element and the nonconforming P nc

1 /P dc0 (Crouzeix/Raviart) Stokes ele-

ment are uniformly “inf-sup” stable. Define the analogues of these Stokes elements on tetrahedralmeshes in 3D and show their uniform “inf-sup” stability.

Exercise 3.11: Show that the “MINI” Stokes element introduced in class is equivalent to aversion of the “stabilized” P c

1 /Pc1 Stokes element with the pressure equation:

(χh,∇ · v(1)h ) +∑

T∈Th

δT (∇χh,∇ph)T =∑

T∈Th

2∑

i=1

δT |T |(1, ϕbT,i)T

(∂iχh, (fi, ϕbT,i)ϕ

bT,i)T ∀χh ∈ Lh,

where v(1)h is the linear part of the MINI-velocity, ϕbT,i the “bulb” basis functions on an element

T and δT ≈ h2T . (Hint: Split the variational equations of the MINI element into equations forthe linear part and its “bulb” component.)

Exercise 3.12: Consider the lowest-order nonconforming P nc1 /P dc

0 Stokes element analyzed in

3.4 Exercises 129

class. Prove for this element the following discrete version of the Poincare inequality on shape-uniform triangulations Th :

‖vh‖ ≤ c‖∇hvh‖, vh ∈ Hh,

where again ∇h denotes the cellwise defined gradient operator.(Hint: One may use a duality argument and the fact that the continuity property of functionsin Hh implies

∫Γ[vh] ds = 0 for Γ ∈ ∂Th, vh ∈ Hh.)

Exercise 3.13: The conforming, “equal order” Stokes element of type P c2 /P

c2 (continuous,

piecewise quadratic velocities and pressures) in its pure form is not “inf-sup” stable. Formulatethe corresponding fully consistent stabilized versions of this Stokes element for approximatinga) the linear Stokes problem and b) the nonlinear Navier-Stokes problem.

Exercise 3.14: Consider the approximation of the nonlinear Navier-Stokes problem as consid-ered in class seeking a v ∈ V = ϕ ∈ H1

0 (Ω)2 |∇ · ϕ = 0 such that

ν(∇v,∇ϕ) + (v · ∇v, ϕ) = (f, ϕ) ∀ϕ ∈ V,

by the finite element method using a conforming “inf-sup” stable Stokes element, e. g., theconforming P c

2/Pdc0 element. Suppose that in this approximation the nonlinear form n(v, v, ϕ) =

(v · ∇v, ϕ) is used in its original nonsymmetrized form, seeking vh ∈ Vh such that

ν(∇vh,∇ϕh) + (vh · ∇vh, ϕh) = (f, ϕh) ∀ϕh ∈ Vh,

In this case, we do not have n(v, ϕ, ϕ) = 0 , which causes difficulties even in proving existenceof discrete solutions.

a) Use the properties of the P c2 /P

dc0 element in 2D to show the estimate

|n(v, ϕ, ϕ)| ≤ c1hL(h)‖∇vh‖‖∇ϕh‖2, vh, ϕh ∈ Vh.

where L(h) := max1, log(1/h) .b) How does the analogue of this estimate in 3D look like?Hint: One may use integration by parts, the definition of the discrete space Vh and the Sobolevinequalities (see Exercise 7.2)

‖wh‖∞ ≤ cL(h)‖∇wh‖ in 2D, and ‖wh‖∞ ≤ ch−1/2‖∇wh‖ in 3D.

Exercise 3.15: Consider the situation described in Exercise 12.3.

a) Apply the Brouwer fixed point theorem (as in the corresponding argument on the continuouslevel) to show that for sufficiently small h ≤ h1 the discrete Navier-Stokes problem is solvable.

b) Prove that for sufficiently small data, c2∗ν−2‖f‖−1 < 1 , the solution obtained in (a) is unique.

Exercise 3.16: Consider again the lowest-order nonconforming P nc1 /P dc

0 Stokes element anal-ized in class. Prove for this element the “discrete” versions of the following Sobolev inequalitieson quasi-uniform triangulations Th :

(i) ‖vh‖6 ≤ cnc∗ ‖∇hvh‖, vh ∈ Hh,

(ii) ‖vh‖3 ≤ cnc∗ ‖vh‖1/2‖∇hvh‖1/2, vh ∈ Hh,


where ∇h denotes again the cellwise defined gradient operator.

(Hints: (i) For proving the first inequality, one may use the following estimate for cellwiseconstant functions w on the quasi-uniform mesh Th in 2D or 3D (Try to give an argument forthis estimate.):

‖w‖6 ≤ ch−1‖w‖.Further, to the (nonconforming) function vh ∈ Hh , one can associate a “smooth” functionv ∈ H by the relation

(∇v,∇ϕ) = (∇hvh,∇ϕ) ∀ϕ ∈ H.Then, considering vh as an approximation to v , by a duality argument, one shows the followingerror estimate (Try to give a complete argument for this estimate.)

‖v − vh‖ ≤ ch‖∇hvh‖.

(ii) The second inequality can be obtained by using the Holder inequality and the first estimate.)

Exercise 3.17: Consider the variational Navier-Stokes problem seeking v ∈ V = ϕ ∈ H =H1

0 (Ω)d |∇ · ϕ = 0 such that

ν(∇v,∇ϕ) + (v · ∇v, ϕ) = (f, ϕ) ∀ϕ ∈ V,

in the case of general data ν, f . Suppose that there exists a solution v ∈ V , for which thederivative form

a′(v;ψ,ϕ) := ν(∇ψ,∇ϕ) + (v · ∇ψ,ϕ) + (ψ · ∇v, ϕ), ψ, ϕ ∈ V,

is coercive, i. e., with some constant α > 0 there holds

supϕ∈V

a′(v;ψ,ϕ)

‖∇ϕ‖ ≥ α‖∇ψ‖, ψ ∈ V.

Show that then this solution is locally unique in some ball BR(v) ⊂ V with radius R = R(α) >0 .

Remark: This concerns the situation, such as in the Taylor problem, in which for the same setof data multiple stationary solutions exist which are only locally unique.

Exercise 3.18: On a convex polygonal domain Ω ⊂ R2 the Laplace operator −∆ (similarly

as the Stokes operator) can be defined as a self-adjoint positive definite operator in L2(Ω)with dense domain of definition D(−∆) = H1

0 (Ω) ∩ H2(Ω) and range R(−∆) = L2(Ω) . Itis invertible with compact inverse −∆−1 : L2(Ω) → L2(Ω) . Therefore, its spectrum σ(−∆)consists of isolated real positive eigenvalues 0 < λ1 ≤ λ2 ≤ λ3 ≤ . . . with no finite accumulationpoint. For any value λ 6∈ σ(−∆) the operator −∆−λI : D(−∆) → L2(Ω) is onto with boundedinverse (−∆ − λI)−1 . The same holds true for the operator −∆ − λI : H1

0 (Ω) → H−1(Ω)(defined in a variational sense). For such λ the bilinear form

aλ(u, ϕ) := (∇u,∇ϕ)− λ(u, ϕ), u, ϕ ∈ V := H10 (Ω),

3.4 Exercises 131

can, in general, not be V -elliptic. Show that it is always “coercive”, i. e.,

supϕ∈V

aλ(u, ϕ)

‖∇ϕ‖ ≥ α‖∇u‖, u ∈ V,

with a constant α > 0 related to the norm of the corresponding operator (−∆ − λI)−1 :H−1(Ω) → H1

0 (Ω) .Hint: You may use an auxiliary problem with appropriate right-hand side and the boundednessof the operator (−∆− λI)−1 : H−1(Ω) → H1

0 (Ω).

Exercise 3.19: Let the singularly perturbed Sturm-Liouville problem

−εu′′ + u′ = 0, in Ω = (0, 1), u(0) = 1, u(1) = 0,

in 1D be discretized by conforming “linear” finite elements on an equidistant mesh 0 = x0 <x1 < · · · < xN < xN+1 = 1 with “artificial diffusion” set to εh := ε + δh for stabilizing thetransport term. Form the corresponding finite difference equation and derive an explicit formulafor its solution. Deduce that the choice δ ≥ 1

2 leads to a diagonally dominant M -matrix.

Exercise 3.20: Consider the discretization of the p-Laplace-like problem (1 < p <∞)

min J(u) :=1

p

∫

Ω(1 + |∇u|2)p/2 dx−

∫

Ωfu dx on V = H1,p

0 (Ω),

in 2D by conforming “linear” finite elements, Vh ⊂ V , on a family of triangulations Thh>0 .Formulate explicitly the corresponding discrete variational equations and the Newton iterationfor their solution.

Exercise 3.21: Let the stationary Navier-Stokes problem with homogeneous Dirchlet boundaryconditions be approximated by a conforming “inf-sup” stable Stokes element leading to the finitedimensional problems: Find vh, ph ∈ Hh × Lh, such that

ν(∇vh,∇ϕh) + n(vh, vh, ϕh)− (ph,∇ · ϕh) = (f, ϕh) ∀ϕh ∈ Hh,

(χh,∇ · vh) = 0 ∀χh ∈ Lh.

Show in analogy to the continuous level with help of the Brouwer fixed point theorem thatthese problems always possess solutions, which are unique if the data are sufficiently small,q = c2∗ν

−2‖f‖−1 < 1.

Exercise 3.22: Let the finite element subspaces Hh × Lh ⊂ H × L be defined on basis ofa conforming Stokes element. Further, suppose that there exists an interpolation operatorπ : h : H → Hh satisfying the relations

(χh,∇ · πhv) = (χh,∇ · v), χh ∈ Lh, v ∈ H,

‖∇πhv‖ ≤ c‖∇v‖, v ∈ H.

Show that then the family of spaces Hh×Lhh>0 is uniformly “inf-sup” stable. Use this resultfor proving the uniform “inf-sup” stability of the conforming P c

1/Pc1 Stokes element (“MINI

element”) in 2D.


Exercise 3.23: Give short answers to the following questions:

1. Which property of a bilinear form a(·, ·) on a Hilbert space H with norm ‖ · ‖H is meantby “H-ellipticity”? Give an example of an H-elliptic bilinear form, which is not symmetric,i. e., not a scalar product.

2. Which properties must a family of triangulations Thh>0 of a polygonal domain Ω ⊂ R2

possess in order to be called “quasi-uniform”?

3. Formulate the “minimal surface problem” on a domain Ω ⊂ R2 . In what sense is here the

boundary condition imposed?

4. Consider the approximation of the Poisson problem on a convex polygonal domain Ω ⊂ R2

by “linear” finite elements on quasi-uniform meshes. What are the best achievable ordersof approximation in the L∞- and the W 1,∞-norm if the sulution satisfies u ∈W 2,∞(Ω) ?

5. What is the content of the Poincare inequality on a bounded domain Ω ⊂ Rd ? Formulate

conditions on the functions considered, under which this inequality holds true.

6. How does the “inf-sup” stability condition for Stokes elements look like and what is itgood for?

7. Give three examples of uniformly “inf-sup” stable pairs of finite element spaces Hh × Lhfor approximating the Stokes problem.

8. How does the functional iteration applied in solving the Navier-Stokes equation on thefunction space level look like? Under what condition is its global convergence guaranteed?

9. What is the purpose of modifying the nonlinear term n(u, v, w) := (u · ∇v,w) in thevariational Navier-Stokes problem to n(u, v, w) := 1

2n(u, v, w) − 12n(u,w, v) ? Show that

n(u, v, w) = n(u, v, w) for functions u ∈ V and v,w ∈ H .

10. What is the purpose of the streamline diffusion stabilization in the finite element ap-proximation of the Navier-Stokes problem? What is the dependence of the stabilizationparameters δT on the local mesh size hT ?

4 Adaptivity (to come)

133

134 Adaptivity (to come)

5 Mixed Methods (to come)

135

136 Mixed Methods (to come)

Bibliography

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[4] R. Rannacher: Numerische Mathematik 3 (Numerische Methoden der Kontinu-umsmechanik), Lecture Notes, Heidelberg University,http://numerik.uni-hd.de/∼lehre/notes/

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