an adaptive surface finite element method for the laplace- …1078803/... · 2017. 3. 6. · degree...

IN DEGREE PROJECT MATHEMATICS,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2017

An Adaptive Surface Finite Element Method for the Laplace-Beltrami Equation

GÖKCE TUBA MASUR

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ENGINEERING SCIENCES

An Adaptive Surface Finite Element Method for the Laplace-Beltrami Equation GÖKCE TUBA MASUR Degree Projects in Scientific Computing (30 ECTS credits) Degree Programme in Applied and Computational Mathematics (120 credits) KTH Royal Institute of Technology year 2017 Supervisor at KTH: Johan Hoffman Examiner at KTH: Michael Hanke

TRITA-MAT-E 2017:05 ISRN-KTH/MAT/E--17/05--SE Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

AbstractIn this thesis, we present an adaptive surface finite elementmethod for the Laplace-Beltrami equation. The equationis known as the manifold equivalent of the Laplace equa-tion. A surface finite element method is formulated for thispartial differential equation which is implemented in FEn-iCS, an open source software project for automated solu-tions of differential equations. We formulate a goal-orientedadaptive mesh refinement method based on a posteriori er-ror estimates which are established with the dual-weightedresidual method. Some computational examples are pro-vided and implementation issues are discussed.

ReferatEn adaptiv finita elementmetod för

Laplace-Beltrami ekvationen

I den här rapporten presenterar vi en adaptiv finite ele-mentmetod för Laplace-Beltrami ekvationen. Ekvationen ärkänd som Laplace ekvation på ytor. En finita elementmetodför ytor formuleras för denna partiella differentialekvationvilken implementeras i FEniCS, en open source mjukva-ra för automatiserad lösning av differentialekvationer. Weformulerar en mål-orienterad adaptiv nätförfinings-metodbaserad på a posteriori feluppskattningar etablerade medhjälp av metoden för dual-viktad residual. Beräkningsex-empel presenteras och implementeringen diskuteras.

.

Acknowledgement.

First, I would like to express my sincere gratitude to my supervisor ProfessorJohan Hoffman for all his valuable guidance and patience throughout this work. Ifeel privileged to be his student and to benefit not only from his inspiring lectureson Finite Element Method and Advanced Computations in Fluid Mechanics but alsofrom his individual assistance. Also, my special thanks goes to the faculty members,N. Cem Değirmenci and Van Dang Nguyen, for their help and guidance, and Dr.Qaisar Latif, Research Associate at Jacobs University Bremen, for his invaluablecomments on the report and help in theoretical understanding. At last but not theleast, feeling the support of my family is always encouraging in every step of mylife.

Contents

List of Figures

1 Introduction 1

2 Background 52.1 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 The Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Weak Formulation of a PDE . . . . . . . . . . . . . . . . . . 72.2.2 The Galerkin FEM . . . . . . . . . . . . . . . . . . . . . . . . 82.2.3 The Finite Element . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Essentials of a Manifold . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Adaptive FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Surface Finite Element Method 153.1 Technical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 The Continuous and Discrete Surfaces . . . . . . . . . . . . . 153.1.2 Lifts and Extensions of Functions . . . . . . . . . . . . . . . . 173.1.3 Parametrized Surfaces . . . . . . . . . . . . . . . . . . . . . . 18

3.2 The Laplace-Beltrami Problem . . . . . . . . . . . . . . . . . . . . . 193.2.1 The Model Problem . . . . . . . . . . . . . . . . . . . . . . . 193.2.2 Green’s Theorem for Manifolds . . . . . . . . . . . . . . . . . 193.2.3 SFEM Formulation . . . . . . . . . . . . . . . . . . . . . . . . 20

4 A Posteriori Error Analysis 234.1 Preparation for Error Analysis . . . . . . . . . . . . . . . . . . . . . 23

4.1.1 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.1.2 Interpolation and Trace Theorems . . . . . . . . . . . . . . . 244.1.3 Relation between Sobolev Spaces . . . . . . . . . . . . . . . . 25

4.2 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2.1 Dual-weighted Residual Method . . . . . . . . . . . . . . . . 274.2.2 L2-error Estimate . . . . . . . . . . . . . . . . . . . . . . . . 284.2.3 Energy Estimate . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Adaptive SFEM 355.1 Numerical Implementaion . . . . . . . . . . . . . . . . . . . . . . . . 355.2 Implementation Results . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2.1 Implementation on a Sphere . . . . . . . . . . . . . . . . . . . 375.2.2 Implementation on a Torus . . . . . . . . . . . . . . . . . . . 41

6 Conclusion and Future Work 456.1 Theoretical Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.2 Implementation Part . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Appendices 47A.1 Laplacian on a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 49A.2 Laplacian on a Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Bibliography 53

List of Figures

3.1 A mesh element T , its projection a(T ), and normals n and nh to Γ andΓh, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 The tubular region Uε around the given surface Γ . . . . . . . . . . . . . 20

5.1 Results for uniform SFEM solving the Laplace-Beltrami equation, −∆Γu =f , on the unit sphere Γ with boundary: a) L2-norm and energy-normerrors, and their theoretical bounds with respect to DOF . b) Residualterm including jump term, and geometric term of the a posterirori errorestimates with respect to DOF . . . . . . . . . . . . . . . . . . . . . . . . 38

5.2 A refined mesh obtained by Adaptive SFEM based on error indicator ηTon the unit sphere, and its magnified view along z-axis. . . . . . . . . . 38

5.3 Adaptive SFEM solutions solving the Laplace-Beltrami equation on theunit sphere at the first and the last steps of the adaptive algorithm. . . 39

5.4 Refined cells with respect to the L2 error bound at the second and the laststeps of the adaptive algorithm. Blue cells are not refined, and red andgreen ones indicate the refined cells where the residual term includingjump over facets, and the geometric terms are dominant, respectively. . 39

5.5 Results for Adaptive SFEM on the unit sphere Γ based on error indicatorηT : a) L2-norm and energy-norm errors, and their bounds with respectto DOF . b) Residual component including the jump, and geometriccomponents of the errors with respect to DOF . . . . . . . . . . . . . . . 40

5.6 Percentage of refined elements whose residual component including thejump term is greater than the geometric component. . . . . . . . . . . . 40

5.7 Results for uniform SFEM solving the Laplace-Beltrami equation, −∆Γu =f , on a torus Γ with major and minor radii 1 and 0.25, and u =exp

(1

1.85−x2

)siny: a) L2-norm and energy-norm errors and their error

bounds with respect to DOF . b) Error bound components with respectto DOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.8 Elements marked for refinement at an intermediate and the last stepsof the adaptive algorithm based on the L2 error bound. Blue elementsare preserved for the next iteration, red and green elements are markedwhere the residual error with the jump term, and the geometric errorare dominant, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.9 Results for Adaptive SFEM on the torus based on error indicator ηT : a)L2 and energy-norm errors with their bounds with respect to DOF . b)Error bound components with respect to DOF . . . . . . . . . . . . . . 42

5.10 Meshes and Adaptive SFEM solutions on the torus. Above: Meshes atthe first and an intermediate steps of the adaptive algorithm. Below:FEM solutions at the first and the last steps of the adaptive algorithm. 43

Chapter 1

Introduction

Problems including partial differential equations (PDEs) over flat domains havebecome a standard tool in numerical analysis. Now, numerical approaches to solvePDEs on curved manifolds have seen a growing interest over the last decades, aproblem which emerges in a wide variety of applications in many engineering andscientific fields. Surface PDEs can be used to model many phenomena in fluidmechanics, material science, image processing and cell-biology. In this report, weconsider the Laplace-Beltrami equation to represent basic issues in theory and im-plementation, and the purpose of this work is to develop an adaptive finite elementmethod (FEM) for this problem. The equation takes the form

−∆Γu = f on Γ,

where Γ is a connected two-dimensional surface embedded in R3, and ∆Γ is theLaplace-Beltrami operator on Γ.

The initial step in constructing a FEM solution to a PDE is to generate a meshfrom the geometric domain, and then to select the finite dimensional space which toseek a solution. Accuracy is limited by the mesh resolution. On the other hand, thecomputation becomes expensive for high number of mesh elements. Thus, adaptivemesh refinement which decreases computational work is desired.

The numerical errors depend on the quality of the mesh, so the mesh generationplays a fundamental role in FEM. To give a good start to a surface finite elementmethod (SFEM), the surface approximation of Γ should consist of “well-shaped"elements. By the term “well-shaped", we mean elements whose aspect-ratio is closeto that of an equilateral triangle. A good start with a high quality mesh is notenough alone, one should provide a good adjustment of the mesh after each adaptiverefinement step to obtain a reliable numerical solution. Adaptive FEM is based ona posteriori error estimation, which provides error indicators for mesh refinementand error bounds that provides stopping criteria for the algorithm.

In SFEM, error bounds may be divided into a “residual indicator term", whicharises from approximating an infinite-dimensional function space by a finite-dimensionalfinite element space, a “geometric error" which arises from replacing Γ with a com-putational mesh, and a “data approximation term" due to the approximation of f

1

CHAPTER 1. INTRODUCTION

on Γ by fh on Γh. The geometric error does not appear in a flat domain but in thepresence of a curved surface, it has the same order as the order of the residual term,for more detail see [8]. A posteriori error analysis in our work will include only theresidual error and the geometric error terms, by omitting the data approximationterm assuming a good approximation to data. The error is estimated for any linearfunctional of the solution using an adjoint based approach, see e.g. [6].

There are some different available methods to approximate Γ. The two mainmethods are: (i) constructing the mesh on a linear approximation of the surface,and (ii) global or local parametrizations. The first approach is considered originallyin Dzuik’s work [11], which is also the first attempt to solve the Laplace-Beltramiequation on a curved surface. In this paper, Γ is represented as a level set of asigned distance function d and is approximated by a polyhedral surface Γh whosefaces are simplices, then piecewise linear finite element functions are used on Γh.In global parametrizations [3], the mesh is directly created on the analytical sur-face. For a sphere, it is achieved by mapping a triangulated planar domain ontothe sphere, and then the projected elements are refined uniformly to have shape-regular curved triangles. The projection of a regular icosahedron, a polyhedronwith twenty equilateral triangle faces, is used as an initial step to approximate asphere, and a piecewise linear finite element approximation is then used. The localparametrizations are based on similar idea.

The linear approximation technique of Γ is used in the theoretical part of thiswork while a global parametrization will be helpful to obtain f from a given func-tion u in the numerical implementation part using manufactured solutions to verifythe method. Although linear approximation creates a geometric error, it has theflexibility to handle approximation of a wide range of surfaces, in particular thosewhich lack parametrizations. Global or local parametrizations do not give any ge-ometric error contribution to the error estimates. Another way to achieve it is toconstruct a finite element space on Γh, and then to take its projection on Γ, whichis introduced in [18]. Also, Dzuik’s method in [11] does not involve any geometricerror.

In this work, we contribute to an Adaptive SFEM for the Laplace-Beltramioperator in theory and in numerical implementation. In theory, we present a newassumption to construct the tubular region Uε of width ε around the surface Γ,and by doing so, in Green’s theorem for manifolds which is presented in [12], wereplace the co-normal vector µ, which is normal to the boundary of Γ, ∂Γ, andtangent to Γ, with a normal vector n to Γ. If the Laplace-Beltrami equation hasnon-homogeneous boundary conditions, we expect that this new result in Green’stheorem will be helpful in both theoretical and implementation parts. Then, wepresent the error bound in L2-norm by the dual-weighted residual method andextend it with the convexity property of Γ, which the L2 error estimator in [8] istaken as a reference. After that, in the same manner that is to use the interpolationtheorems in Sobolev spaces, the energy-norm error bound is established. In thenumerical implementation part, we observe that the do-nothing method also worksefficiently as an adaptive method.

2

The structure of the thesis is as follows:

Chapter 2 Adaptive finite element methods are introduced with function spaces.

Chapter 3 SFEM and its technical preliminaries are provided and after intro-ducing the model problem, the surface finite element method of theproblem is stated.

Chapter 4 Interpolation error theorems are introduced, and then an a posteriorierror estimate with respect to goal functional is proven

Chapter 5 The adaptive algorithm based on the error estimate is presented, andthe numerical implementation is discussed.

Chapter 6 A brief conclusion is presented and possible future work is discussed.

3

Chapter 2

Background

The purpose of this chapter is to present the background of adaptive FEM. TheFEM solution of any differential equation depends on the function space, so we alsoreview some basics of the function spaces that are important for our work.

2.1 Function SpacesLet Ω be a Lebesgue measurable subset of Rn with non-empty interior. We

recall some basic facts from [5] and [13].

Definition 2.1.1. Let f be a real-valued Lebesgue measurable function on Ω. Letus fix the following notation for the norm: for 1 ≤ p <∞,

||f ||Lp(Ω) :=(∫

Ω|f(x)|pdx

)1/p,

and for the case p =∞,

||f ||Lp(Ω) := ess sup|f(x) | x ∈ Ω.

The Lebesgue space Lp(Ω) is defined as

Lp(Ω) := f | ||f ||Lp(Ω) <∞.

Remark 2.1.2. In FEM formulation of a PDE, which can be seen later, the point-wise values of derivatives are not needed, only derivatives that can be interpreted asfunctions in Lp(Ω) occur.

Definition 2.1.3. The set of locally integrable functions on Ω is denoted by

L1loc(Ω) := f | f ∈ L1(K), ∀ compact K ⊂ interior Ω.

The definition of locally integrable functions gives flexibility to functions inL1loc(Ω) behave arbitrarily badly near the boundary, and for these functions, we

define a weak derivative.

5

CHAPTER 2. BACKGROUND

Definition 2.1.4. Let f ∈ L1loc(Ω). A function g ∈ L1

loc(Ω) is a weak derivative oforder α of f , denoted Dα

w(f), if∫Ωg(x)φ(x)dx = (−1)|α|

∫Ωf(x)φ(α)(x)dx ∀ φ ∈ C∞0 (Ω),

where C∞0 (Ω) is the set of infinitely differentiable functions with compact supportin Ω, and φ(α) is the α order derivative of φ.

Definition 2.1.5. Let k be a non-negative integer. The Sobolev norm of f ∈ L1loc(Ω)

is defined by for 1 ≤ p <∞,

||f ||Wkp (Ω) :=

( ∑|α|≤k

||Dαwf ||

pLp(Ω)

)1/p, (2.1.1)

and for p =∞,||f ||Wk

∞(Ω) := max|α|≤k

||Dαwf ||L∞(Ω).

The Sobolev space W kp is

W kp (Ω) := f ∈ L1

loc(Ω) | ||f ||Wkp (Ω) <∞. (2.1.2)

Definition 2.1.6. For k a non-negative integer and f ∈W kp (Ω), the Sobolev semi-

norm is defined by

|f |Wkp (Ω) :=

( ∑|α|=k

||Dαwf ||

pLp(Ω)

)1/p, (2.1.3)

in the case 1 ≤ p <∞, and in the case p =∞,

|f |Wk∞(Ω) := max

|α|=k||Dα

wf ||L∞(Ω).

Proposition 2.1.7. Let Ω be a domain, and k and m be non-negative integers suchthat k ≤ m. If 1 ≤ p ≤ ∞ ,then

Wmp (Ω) ⊂W k

p (Ω).

If p and q are real numbers such that 1 ≤ p ≤ q ≤ ∞, then

W kq (Ω) ⊂W k

p (Ω).

Remark 2.1.8. For k ≥ 0 and p = 2, there is an inherited Hilbert space structureon the Sobolev space W k

2 (Ω) with an inner-product given by

(u, v)Hk(Ω) =∑|α|≤k

(Dαwu,D

αwu)L2(Ω).

In this case, we use the following notation

Hk(Ω) := W k2 (Ω). (2.1.4)

6

2.2. THE FINITE ELEMENT METHOD

Definition 2.1.9. Let Hk0 (Ω) be the closure of C∞0 (Ω) in Hk(Ω). The space of

bounded linear maps f : Hk0 (Ω) → R is denoted by H−k(Ω) = Hk

0 (Ω)∗, the dualspace of Hk

0 (Ω), and the action of f ∈ H−k(Ω) on v ∈ H−k(Ω) by 〈f, v〉. Then, thenorm of f ∈ H−k(Ω) is given by

||f ||H−k(Ω) := supv∈Hk

0 (Ω),v 6=0

〈f, v〉||v||Hk(Ω)

. (2.1.5)

2.2 The Finite Element MethodIn this section, we present a weak formulation of an abstract PDE, the Galerkin

FEM, and the definition of finite element with its implementation, for more details,see [5] and [13].

2.2.1 Weak Formulation of a PDEFEM is an important numerical approach to approximate the solutions of dif-

ferential equations in a finite dimensional space. Let Ω be a domain in Rn, then weconsider PDEs of the form

Au = f on Ω, (2.2.1)where A is a linear differential operator and f is a given data. The problem in(2.2.1) is called a strong form of the differential equation.

We refer to the set of functions where we seek the numerical solution to (2.2.1)as the trial space, denoted from now on by S, which for some A may be seen as anatural interpretation of a Hilbert space with an inner product a(., .), i.e.,

S := v ∈ L2(Ω) | a(v, v) ≤ ∞. (2.2.2)

The weak or variational formulation of the problem is obtained by multiplying thestrong form with a test function v, and then integrating over the domain, in otherwords we seek a solution to the following equation,

a(u, v) := 〈Au, v〉 = 〈f, v〉 =: L(v) ∀ v ∈ V, (2.2.3)

where V is the space of test functions v, and if A = −∆ then A is a linear mappingfrom H1(Rn) to H−1(Rn). Let us notice that a and L in (2.2.3) are respectively abilinear form and a linear form. Assuming sufficient regularity of the weak solution,it is also a strong solution. The following theorem can be proven for the Laplaceequation with A = −∆ [5].

Theorem 2.2.1. A C2(Ω) solution to the weak form of FEM is a also a solutionto the corresponding strong form.

In dealing with a boundary value problem, the Neumann boundary conditionsappear in the weak formulation, whereas the Dirichlet boundary conditions areenforced through the definition of the space, and therefore, affect the correspondingweak formulation implicitly.

7


2.2.2 The Galerkin FEM

In the Galerkin FEM, the weak formulation (2.2.3) is discretized on a mesh Thof Ω, that is a union of a finite number M of compact and connected Lipschitzdomains Ti with non-empty interior such that TiMi=1 forms a partition of Ω. LetSh ⊂ S and Vh ⊂ V be finite dimensional subspaces in the form of piecewise linearfunctions defined over the mesh elements T of Th and continuous on Ω, i.e.,

Vh := vh ∈ C(Ω) | vh|T ∈ P1(T ) for T ∈ Th, (2.2.4)

where P1(T ) is the set of linear functions on T . Sh and Vh are known as finiteelement spaces of trial functions and test functions, respectively. We seek a solutionuh ∈ Sh over the test functions in Vh, in other words, find uh ∈ Sh such that

a(uh, vh) = (f, vh) ∀ vh ∈ Vh. (2.2.5)

For A = −∆, the following theorem can be seen in [5].

Theorem 2.2.2. Let f ∈ L2(Ω). The discretized weak formulation, (2.2.5), has aunique solution.

Notice the following relation which is known as the Galerkin orthogonality be-tween the weak solution and the discrete FEM solution

a(u− uh, vh) = 0 ∀ vh ∈ Vh. (2.2.6)

Thus, the approximate solution is determined as the member of a finite-dimensionalspace Sh for which the residual error is orthogonal to a set of test functions vh, andif the differential operator A admits the definition of an energy-norm by

||v||E =√a(v, v),

then the FEM solution uh is optimal with respect to this norm, which is proven in[5].

2.2.3 The Finite Element

We now define a finite element in Rn following [5] and the FEM method for themodel problem will follow.

Definition 2.2.1. A finite element is a triplet (T,P,N ), where

(i) T ⊂ Rn is a bounded closed set with nonempty interior and piecewise smoothboundary (the element domain),

(ii) P is a finite dimensional space of functions on T (the space of shape functions),and

8

2.2. THE FINITE ELEMENT METHOD

(iii) N = N1, ...Nk is a basis for the dual space P ′ (the set of nodal variables orthe dual basis).

Definition 2.2.2. Let (T,P,N ) be a finite element. The basis φ1, φ2, ..., φk ofP dual to N (i.e., Ni(φj) = δij where δij takes values 1 and 0 for i = j and i 6= j,respectively) is called the nodal basis of P.

After introducing the Galerkin method and Definition 2.2.1, the finite elementmethod will be formulated for the model problem

−∆u = f on Ωu = 0 in ∂Ω

(2.2.7)

where ∂Ω is the boundary of the domain Ω. Bilinear and linear forms are

a(u, v) = (−∆u, v)Ω =(∇u,∇v)Ω − (~n∇u, v)∂Ω = (∇u,∇v)Ω

L(v) =(f, v)Ω(2.2.8)

where the first equality in (2.2.8) is obtained by using Green’s theorem; and thesubscripts denote the domain of integration in the L2-inner product. We chooseS = V to be the Hilbert space H1

0 (Ω), the set of functions in H1(Ω) with a vanishingtrace on ∂Ω.

To formulate the Galerkin FEM method, take Vh = vh ∈ C(Ω) | vh|T ∈P1(T ), v|∂Ω = 0, which is a Hilbert space H1 on Th satisfying the Dirichlet bound-ary condition, and then seek uh ∈ Vh such that

a(uh, vh) =(∇uh,∇vh)Ωh= (f, vh)Ωh

= L(vh), ∀ vh ∈ Vh. (2.2.9)

It is possible to write the approximated solution in the basis of Vh. For a meshelement T , let zi and λi ∈ P1(T ), i = 1, 2, 3, be respectively the nodes, and theelement basis functions of T where

λi(zj) = δij . (2.2.10)

It is important to have a nodal basis φj for Vh, where the element basis functionson neighboring elements should have the same values on the nodes of the commonedges using the continuity requirement. The set of nodal basis functions φjkj=1,where k is the number of the nodes in Th, is also referred to as a set of tent functions.For each neighboring triangle T of zj , the relation between the element basis andthe nodal basis on a node aj can be written as

φi(zj) = λT,l(zT,m) = δlm 1 ≤ l,m ≤ 3, (2.2.11)

where λT,l is the element basis function, described in (2.2.10), on T with localnumbering l, and zT,m is local numbering of zj in the cell T . Now, the tent functionsare a basis for Vh, and for any uh ∈ Vh, we have the representation

uh(x) =k∑i=1

uh(zi)φi(x), (2.2.12)

9


where Ui = uh(zi) = Ni(uh) for 1 ≤ i ≤ k are unknown coefficients known as thedegrees of freedom of the system [13]. The test function vh is represented by thebasis functions φj for 1 ≤ j ≤ k, and (2.2.9) takes the form: find Ui for 1 ≤ i ≤ k,such that

k∑i=1

Ui(∇φi∇φj)Ω = (f, φj)Ω 1 ≤ j ≤ k, (2.2.13)

which can be expressed in the matrix form

AU = F,

[A]i,j = (∇φi∇φj)Ω,

[F ]j = (f, φj)Ω,

[U ]i = Ui,

(2.2.14)

where the matrix A is a stiffness matrix and the vector F is a load vector.Let us recall the definition of an affine equivalent finite element from [5].

Definition 2.2.3. Let (T,P,N ) be a finite element and let A(x) = Ax + b be anaffine map, where A is a nonsingular matrix. A finite element (T , P, N ) is affineequivalent to (T,P,N ) if

(i) A(T ) = T ,

(ii) A∗(P) = P, and

(iii) A∗N = N ,

where the pull-back A∗ is defined by A∗(f) := f A and the push-forward A∗ isdefined by (A∗N)(f) := N(A∗(f)) for a function f ∈ P.

In view of this definition, for each element T in the mesh, it is possible totake a triangle whose corners are at the points (0, 0), (0, 1) and (1, 0) as an affineequivalent element A(T ) or a reference element. The functions defined on A(T ) areused instead of explicit computation with the basis functions φi of Vh, then by ause of change of variables, the matrices A and F are computed. More about affinetransformation can be found in [8].

2.3 Essentials of a ManifoldFor our consideration, we need mostly one or two dimensional manifolds, there-

fore we recall the basics for them; but however, the notion exists in full generalityand for further reading, we refer to [15].

Definition 2.3.1. Let U and V be subsets of Rm and Rn, respectively. If a mapf : U → V is continuous and bijective, and if its inverse map f−1 : V → U isalso continuous, then f is called a homeomorphism and U and V are said to behomeomorphic.

10

2.3. ESSENTIALS OF A MANIFOLD

Definition 2.3.2. A topological space Γ is a locally Euclidean of dimension n ifevery point p in Γ has a neighborhood U such that there is a homeomorphism ϕfrom U onto an open subset of Rn. The pair (U,ϕ : U → Rn) is called a chart at p.

Definition 2.3.3. Let a topological space Γ be a n-dimensional manifold, and(Uα, ϕα) and (Uβ, ϕβ) be its two charts. The composite of homeomorphisms, ϕαand ϕβ,

ϕβ ϕ−1α : ϕ1(Uα ∩ Uβ)→ ϕβ(Uα ∩ Uβ) (2.3.1)

is called transition maps, and if the transition maps are smooth then Γ is a smoothmanifold.

Let us assume from now on that Γ is a 2-dimensional smooth manifold.

Definition 2.3.4. Consider a chart (Uα, ϕα) at a point p ∈ Γ, say ϕα(p) = (x1, x2).The tangent space TpΓ at p is a linear space spanned by the derivations

∂

∂x1|p and ∂

∂x2|p, (2.3.2)

where for any smooth f : Uα → R,

∂

∂xj|pf = ∂

∂xj|p(f ϕ−1

α

)(ϕα(p)) , j = 1, 2. (2.3.3)

Definition 2.3.5. The tangent bundle TΓ of a manifold Γ is the disjoint union oftangent spaces TpΓ at all points of Γ, i.e.,

TΓ :=⋃p∈Γ

TpΓ.

Definition 2.3.6. A vector field X on an open subset U of Γ is a smooth functionthat assigns to each point p ∈ U a tangent vector Xp ∈ TpΓ,

X : U → TΓp 7→ Xp ∈ TpΓ.

The tangent space TpΓ has basis ∂/∂xj |p, the vector Xp is a linear combination

Xp =∑

aj(p)∂

∂xj|p, p ∈ U, aj(p) ∈ R.

Definition 2.3.7. The dual space T ∗pΓ to the tangent space TpΓ at the point p of amanifold Γ is called cotangent space.

Definition 2.3.8. The underlying set of the cotangent bundle T ∗Γ of a manifold Γis the disjoint union of cotangent spaces at all the points of Γ, that is

T ∗Γ :=⋃p∈Γ

T ∗pΓ.

11


Definition 2.3.9. A covector field or differential 1-form on an open subset U of Γis a smooth function ω that assigns to each point p ∈ U a covector ωp ∈ T ∗pΓ,

ω : U → T ∗Γp 7→ ωp ∈ T ∗pΓ.

From any smooth function f : U → R, the differential 1-form of f is constructed asfollows: for p ∈ U and Xp ∈ TpU , define

(df)p (Xp) = Xpf.

Definition 2.3.10. Let f and g be two alternating multilinear functions on a vectorspace V . Then the wedge product, also called exterior product, is defined as follows:for f ∈ Ak(V ) and g ∈ A`(V )

f ∧ g = 1k!`!A(f ⊗ g).

where ⊗ is tensor product.

Definition 2.3.11. A differential form ω of degree 2 on an open subset U of Γ isa function that assigns to each point p ∈ U a alternating 2-linear function on thetangent space TpΓ, ωp ∈ A2(TpΓ),

ω : U → T ∗Γ ∧ T ∗Γp 7→ ωp ∈ T ∗pΓ ∧ T ∗pΓ = A2(TpΓ).

2.4 Adaptive FEMWe remark that Th is an initial mesh, and u and uh are respectively an exact

and an approximate solution to (2.2.7). We want to design an adaptive algorithmbased on an a posteriori error estimate to optimize the computational work to reacha certain accuracy.

A posteriori error estimates gathers information from the problem data and anumerical solution to evaluate an error bound in a given norm ||.|| . As describedin [14], the error has three sources including a Galerkin discretization, quadrature,and solution of the discrete problem, in other words

||u− uh|| ≤ E(uh, h, f), (2.4.1)

where h is the maximum of element diameters, that is the diameter of the smallestball containing a mesh element T . The Galerkin discretization error arises out ofapproximating the solution by piecewise polynomials. Numerical quadrature is usedto evaluate the integrals in the discrete system, which causes the quadrature error.The last error results from solving the resulting discrete systems only approximately,for example, using Newton’s method.

12

2.4. ADAPTIVE FEM

The computational error in (2.4.1) may be written as a representation of theerror in terms of the residual of the approximate solution and the solution to acontinuous dual or adjoint problem. After use of the Galerkin orthogonality, localinterpolation estimates for the dual solution follow, and strong-stability estimatesfor the continuous dual problem lead us to an a posteriori error estimate of theproblem on the current mesh [14].

In the adaptive algorithm, we let a tolerance TOL be a positive value, and wewant to find a mesh Th that requires the least amount of computational work toachieve

||u− uh|| ≤ TOL.

Reliability in the adaptive method can be achieved satisfying the given tolerance byrefinement of the mesh Th. Also, efficiency can be achieved by obtaining an approx-imate solution in the specified tolerance without having unnecessary refinement inthe mesh Th.

To meet the desired accuracy in the algorithm, we seek a corresponding finiteelement approximation uh, with a minimal number of the degrees of freedom, thatsatisfy the following stopping criterion

E(uh, h, f) ≤ TOL. (2.4.2)

If the stopping criterion is satisfied, then the approximate solution uh is accepted.On the other hand, if the stopping criterion is not satisfied, then the mesh is refined,and the discrete system is solved on the new mesh [13].

13

Chapter 3

Surface Finite Element Method

In this chapter, we review linear approximation of surfaces, relations betweencontinuous and discrete surfaces, and parametric representation formulas for asphere and a torus.

3.1 Technical PreliminariesInitially, functions and projections are described on the continuous surface Γ,

and the discrete surface Γh. The surface derivatives, its projections and matrices,and the parametric representation will follow in order.

3.1.1 The Continuous and Discrete Surfaces

Let Γ be a smooth, compact, connected and oriented surface embedded in R3

and Γh be polyhedron approximation of Γ consisting of a mesh Th with triangularmesh elements T . By setting the surface Γ as the zero level set, we define a signeddistance function d(x) by

|d(x)| = dist(x,Γ)

on some open subset Uo of R3. For a sphere centered at xc with radius r, thedistance function is

d(x) = ||x− xc|| − r ∀ x ∈ R3,

where ||x||2 = x2 + y2 + z2. Γ can be represented by the implicit function d asfollows

Γ = x ∈ Uo | d(x) = 0,

where d(x) = 0 refers to x being on the specified surface Γ. For x in interior of Γ,d(x) < 0, and for x in exterior of Γ, d(x) > 0. The outward unit normal vector onΓ can be computed from gradient of d(x) as

~n = ∇d(x)|∇d(x)| ,

15

CHAPTER 3. SURFACE FINITE ELEMENT METHOD

a(T )

T

~n

~nh

Figure 3.1: A mesh element T , its projection a(T ), and normals n and nh to Γ andΓh, respectively.

while the unit normal vector on simplical faces T of Γh is denoted by ~nh.In order to compare functions separately defined on a continuous and a discrete

surface, we need to ‘move’ one of the functions to the surface on which we desirecomparison. To achieve this, we need a projection which maps a point x froma tubular region Uε of width ε about Γ to a closest point on Γ. We define theprojection of a point x ∈ R3 by

a(x) := x− d(x)~n for x ∈ Uε.

The tubular region Uε ⊂ R3, with a sufficiently small width ε, ensures the uniquenessof x ∈ Uε such that x = a(x) + d(x)~n. We may then extend a function v defined onΓ to Uε by

v`(x) = v(a(x)) for x ∈ Uε. (3.1.1)

By the projection a, the surface Γ is approximated with mesh elements projectionsa(T ), see Figure 3.1, then Γ is approximately represented by the union of a(Ti)Mi=1for M finite number of domains.

The Hessian map H : R3 → R3×3 is then defined by H(x) = D2d(x) on Γ. Theprojection matrix onto the tangent space of Γ is

P (x) = I − ~n⊗ ~n,

where ⊗ is the tensor product. By inserting ~nh instead of ~n, we get the projectionmatrix Ph(x) onto that of Γh.

The tangential gradient on Γ is defined by

∇Γv(x) = ∇v(x)− (~n · ∇v(x))~n = P∇v(x)

for a function v defined on Uε and x ∈ Γ. The tangential gradient can also bewritten as

∇Γv(x) = [D1v(x), ..., Dnv(x)],

16

3.1. TECHNICAL PRELIMINARIES

whereDi = ∂

∂xi()− ni

n∑j

∂

∂xj()nj . (3.1.2)

If v was defined on Uε and x ∈ Γh, the tangential derivative on Γh would be

∇Γhv(x) = ∇v(x)− (~nh · ∇v(x))~nh = Ph∇v(x)

on each triangle T with unit normal ~nh. If v is defined on Γ for x ∈ Γ, and vh isdefined on Γh for x ∈ Γh, then their tangential derivatives are defined as following:

∇Γv(x) = P∇v`(x) for x ∈ Γ

and∇Γh

vh(x) = Ph∇v`h(x) for x ∈ Γh,

respectively. Finally, we define the Laplace-Beltrami operator as

−∆Γ = −∇Γ · ∇Γ.

3.1.2 Lifts and Extensions of FunctionsLet vh be a given function defined on Γh. A lift vh on Γ is defined by

vh(x) = vh(a(x)) for x ∈ Γh.

The domain of vh is extended to Uε as

v`h(x) = vh(a(x)) for x ∈ Uε,

by (3.1.1). Thus, we attain v`h as the lift of vh; the lift v`h includes a lifting oper-ation followed by an extension of the domain. Let µh(x) be the Jacobian of thetransformation a(x) such that

µh(x)dσh(x) = dσ(a(x)) for x ∈ Γh, (3.1.3)

where dσ and dσh are differential forms on Γ and Γh, respectively.Our notation is consistent with Demlow and Dzuik’s work [10]. Let us recall the

following facts, stated here as lemmas, from [10].

Lemma 3.1.1. Let vh be defined on Γh, and x ∈ Γh. The relation between∇Γv

`h(a(x)) and ∇Γh

vh(x) is given by

∇Γv`h(a(x)) = [(I − d(x)H(x))]−1

[I − ~nh ⊗ ~n

~nh · ~n

]∇Γh

vh(x). (3.1.4)

For notational ease, we drop the arrow in the normals n and nh, and x infunctions and projections, and dH means d(x)H(x) without any confusion with thedifferentiation sign.

17


Lemma 3.1.2. Let vh be defined on Γh, and x ∈ Γh. For ψ ∈ H1(Γ), we get∫Γh

∇Γhvh(x)∇Γh

ψh(x)dσh(x) =∫

ΓAh(x)∇Γh

v`h(a(x))∇Γhψ`h(a(x))dσ(a(x)),

(3.1.5)where

Ah(x) = 1µh(x) [P (I − dH)Ph(I − dH)P ](x). (3.1.6)

3.1.3 Parametrized SurfacesFor theoretical reasons, an implicit representation of a surface is important for

us, nevertheless a parametric presentation is equally important to find the sourcefunction f from the exact solution in the Laplace-Beltrami equation.

Let Γ be a manifold and g be its Riemeannian metric presented in parametriccoordinates, i.e.

g = gijdxidxj ,

where gij = g(∂i, ∂j), and the inverse of matrix [gij ] is denoted by [gij ]. It followsthat the Laplacian on a Riemann manifold is given in [15, p.93] by

∆Γ = 1√|det gij |

n∑i,j=1

∂

∂xi

(gij√|det gij |

∂

∂xj

). (3.1.7)

Now, we can derive the Laplacian for a sphere and a torus. Let Γ be a spherewith angular spherical coordinates (θ, φ) with parametric representation

(x, y, z) = (rsinθcosφ, rsinθsinφ, rcosθ), (3.1.8)

and the metricg = r2dθ2 + r2sin2θdφ2,

where φ, (0 ≤ φ < 2π), is the azimuthal angle in the xy-plane and θ, (0 ≤ θ ≤ π),is the polar angle from z-axis. Then, the Laplacian on a sphere derived by (3.1.7)is

∆Γ = 1r2sinθ

∂

∂θ

(sinθ ∂

∂θ

)+ 1r2sin2θ

∂2

∂φ2 . (3.1.9)

Let Γ be a torus which has major radius R and minor radius r with the para-metric representation

(x, y, z) = ((R+ rcosθ)cosφ, (R+ rcosθ)sinφ, rsinθ) ,

and the metricg = r2dθ2 + (R+ rcosθ)2dφ2, (3.1.10)

where φ is the azimuthal angle on xy-plane, θ is the altitude angle at z-axis and φ,θ ∈ [0, 2π]. Then, the laplacian on a torus by (3.1.7) is

∆Γ = 1r2(R+ rcosθ)

∂

∂θ

((R+ rcosθ) ∂

∂θ

)+ 1

(R+ rcosθ)2∂2

∂φ2 . (3.1.11)

18

3.2. THE LAPLACE-BELTRAMI PROBLEM

3.2 The Laplace-Beltrami ProblemIn this section, we provide the model Laplace-Beltrami equation, and formulate

the SFEM for the problem.

3.2.1 The Model ProblemLet us recall the Laplace-Beltrami equation. Let Γ be a compact C2 hypersurface

embedded in R3. For a given source f , we seek u to solve the problem

−∆Γu = f on Γ. (3.2.1)

There is a vanishing trace of the boundary if ∂Γ 6= ∅. On the other hand, if ∂Γ = ∅,then there is a unique solution, up to a constant; in particular, the compatibilityconditions

∫Γ udσ =

∫Γ fdσ = 0 determine u uniquely or in other words, determine

the constant.

3.2.2 Green’s Theorem for ManifoldsLet us reconsider Green’s theorem and its proof for a manifold Γ from [12,

Theorem 2.14, p.301].

Theorem 3.2.1. Let Γ be a C2 hypersurface in R3 with smooth boundary ∂Γ. Forf ∈ C1(Γ) and g ∈ C2(Γ), we get∫

Γ(∇Γf · ∇Γg)dσ = −

∫Γf∆gdσ +

∫∂Γ

(f∇Γg · µ)ds, (3.2.2)

where µ is the co-normal vector, that is normal to ∂Γ and tangent to Γ, and dσ andds are respectively the differential 2-form and the differential 1-form.

Let Uε be a tubular region around the surface Γ with width ε such that

∂Uε = Γ(ε) ∪ Γ(−ε) ∪M(ε),

whereΓ(ε) = x | d(x) = ε,Γ(−ε) = x | d(x) = −ε,M(ε) = x+ r~nΓ(x) | x ∈ ∂Γ, r ∈ [−ε, ε],

with ~nΓ being a unit normal vector to Γ, see Figure 3.2, where M(ε) is the redsurface. In the proof of [12, Theorem 2.14, p.301], M(ε) is taken to be a surfaceorthogonal to Γ; however, we consider M(ε) as follows:

LetM(ε) have curvature and define it to be a semicircle for each boundary pointx∂Γ. For a fixed boundary point x∂Γ, let x be an axis such that d(x) = |x−x∂Γ| ∈ xfor x ∈ R3. For d(x) > 0, which means x ∈ Γ(ε), it is +x direction, and for d(x) < 0,which means x ∈ Γ(−ε), it is −x direction. Also, let y be an axis that is tangent to

19


UεM(ε)

M(ε)Γ

Γ(ε)

Γ(−ε)

+x

θ

+y−x

µ

n

Figure 3.2: The tubular region Uε around the given surface Γand parallel surfaces with Γ(ε), Γ(−ε), and normal nΓ to Γ and normal µ to M(ε).

Γ on ∂Γ, and the +y direction is outward from ∂Γ. Then, we define an azimuthalangle θ on the xy-plane, which starts at the +x-axis with value 0 towards the +ydirection and ends at the − x-axis with value π. Finally, we can define M(ε) for afixed x∂Γ as follows:

M(ε) = x | (xcosθ)2 + (xsinθ)2 = ε2, 0 ≤ θ ≤ π,

and according to our new assumption, the description of µ(x) is updated as a normalvector on the curved surfaceM(ε). For a sufficiently small value ε, M(ε) gets closerto x∂Γ, and µ approaches to nΓ at x∂Γ. In [12], f(x) is defined as f(a(x)), andas d(x) gets closer to 0, it follows from the definition of a(x) that f(x) = f(x),therefore

limε→0

∫M(ε)

f(x)µM(ε)(x)dσ(x) =∫∂Γf(x)nΓ(x)ds(x).

In view of the above equation, (3.2.2) changes into∫Γ(∇Γf · ∇Γg)dσ = −

∫Γf∆gdσ +

∫∂Γ

(f∇Γg · nΓ)ds. (3.2.3)

Remark 3.2.1. In the formulation of the model problem, we will observe that thereis no boundary term that appears in the weak formulation; however, in the presenceof a non-homogeneous boundary condition, this result may help not only in thetheory, but also, in the implementation part by avoiding the hardship of defining atangent to Γ on ∂Γ.

3.2.3 SFEM FormulationThe model problem in (3.2.1) is multiplied with a test function v ∈ V followed

by integration over Γ, and after applying Green’s theorem in (3.2.3), we obtain thebilinear form as follows:

20

3.2. THE LAPLACE-BELTRAMI PROBLEM

−∫

Γ(∆Γu)vdσ =

∫Γ(∇Γu · ∇Γv)dσ −

∫∂Γ

(v∇Γu · n)ds ∀ v ∈ V.

For manifolds with boundary, the test and trial function spaces are equal toeach other, S = V = v ∈ H1(Γ) | v|∂Γ = 0, which is also denoted by H1

0 (Γ), andfor manifolds without boundary, the trial space is S = v ∈ H1(Γ) |

∫Γ vdσ = 0,

and V = H1(Γ). We note that the Hilbert spaces, Hk = W k2 , on a surface Γ is

produced by replacing the weak derivatives Dαw in (2.1.1) with the tangential weak

derivatives Dαi , see [12, p.299]. Then, in the weak formulation, we seek a solution

u ∈ S for a given f ∈ L2(Γ) such that∫Γ(∇Γu · ∇Γv)dσ =

∫Γfvdσ ∀ v ∈ V. (3.2.4)

Let us recall the following theorem from [11], which guarantees a solution to theweak form.

Theorem 3.2.2. Let Γ ∈ C2. If ∂Γ 6= ∅, for every f ∈ L2(Γ), for Γ be a convexdomain, there exist a unique weak solution u ∈ H1

0 (Γ) of the problem (3.2.4) ∀v ∈H1

0 (Γ) and||u||H2(Γ) ≤ C||f ||L2(Γ).

If ∂Γ = ∅, for every f ∈ L2(Γ) satisfying∫

Γ fdσ = 0, there exist a weak solutionu ∈ H1(Γ) of the problem (3.2.4) ∀u ∈ H1(Γ), which u is unique up to a constantand

||u||H2(Γ) ≤ C[||f ||L2(Γ) + ||u||L2(Γ)

]. (3.2.5)

Notice that the Laplace-Beltrami equation has a compatibility condition, i.e.||u||L1(Γ) = 0 on manifolds without boundary, and due to the relation ||u||L2(Γ) ≤||u||L1(Γ), the last term in (3.2.5) vanishes such that we can write the followingcorollary.

Corollary 3.2.3. Under the conditions of Theorem 3.2.2 with a compatibility con-dition

∫Γ udσ = 0 if ∂Γ = ∅, the regularity condition, for Γ a convex domain, is

||u||H2(Γ) ≤ C||f ||L2(Γ). (3.2.6)

Let Γh be a polyhedral approximation to Γ, and Sh ⊂ S and Vh ⊂ V are thefinite element spaces of piecewise linear functions defined over the faces of Γh. Wedenote fh to be an approximation to f on Γh. Then, the discrete SFEM problemfollows: find uh ∈ Sh that solves the equation∫

Γh

(∇Γhuh · ∇Γh

vh)dσh =∫

Γh

fhvhdσh ∀ vh ∈ Vh. (3.2.7)

21

Chapter 4

A Posteriori Error Analysis

We devote this chapter to a residual type error analysis of the Laplace-Beltramioperator. Theorems and some inequalities, which will be used in error analysis, arereviewed, and then the a posteriori error analysis is presented.

4.1 Preparation for Error AnalysisIn this section, after the notion of interpolation is introduced, we recall the

interpolation theorems, relation between Sobolev norms defined on the discrete andcontinuous surfaces, and other useful inequalities.

4.1.1 InterpolationWe recall the following definitions from [5].

Definition 4.1.1. Let (T,P,N ), in Definition 2.2.1, be a finite element, and theset φi : 1 ≤ i ≤ k ⊂ P be the dual basis to N . If v is a function for which allNi ∈ N , i = 1, ..., k, then we define the local interpolant by

IT v :=k∑i=1

Ni(v)φi. (4.1.1)

Definition 4.1.2. Let Ω be a domain and Th, 0 < h ≤ 1, be a family of subdivi-sions such that

maxhT : T ∈ Th ≤ h diam Ω, (4.1.2)where hT = diam T . The family is said to be non-degenerate or shape regular if theratio hT /hBT

is uniformly bounded for all T ∈ Th and h ∈ (0, 1], where hBTis the

diameter of the largest ball BT contained in T .

The main idea in residual type error estimates is the use of the continuous dualproblem, and then interpolating the continuous dual solution. We apply the Scott-Zhang type interpolation operator to estimate the continuous dual solution by alocal average of the underlying finite element space.

23

CHAPTER 4. A POSTERIORI ERROR ANALYSIS

In the interpolation operator, it is assumed that n−dimensional meshes, n ≥ 2,are shape regular, and homogeneous boundary conditions are satisfied naturally.This operator averages the interpolated functions on the domain of each nodalfunction value, and it is of optimal order in approximation. The Scott-Zhang typelocal interpolant is denoted by Ih, and maps a function v ∈ W k

p (Ω) to the finiteelement space Sh(Ω) by

Ih(x) =K∑i=1

(∫Ti

ϕi(η)v(η)dη)φi(x), (4.1.3)

where K is the number of nodal points in Th, Ti is a subset of a particular domainTi including the node zi, and φi and ϕi are respectively the basis and the dual basisfor Ti, see [5] and [11] for details.

4.1.2 Interpolation and Trace TheoremsThe interpolation theorems are recalled from [5, Chapter 4], and more informa-

tion including proofs of theorems can be also found in [17].

Definition 4.1.3. A domain T is said to be star-shaped with respect to a ball B iffor all x ∈ T , the closed convex hull of x ∪B is a subset of T .

Theorem 4.1.1. Let (T,P,N ) be a finite element that satisfy the following prop-erties:

(i) T is star shaped with respect to some ball,

(ii) Pm−1 ⊆ P ⊆ Wm∞(T ), where Pm is a set of polynomials in n variables of

degree less than or equal to m, and

(iii) N ⊆(C l(T )

)′, ie. the nodal variables in N involve derivatives up to order l.

If 1 ≤ p ≤ ∞ and either m− l− n/p > 0 when p > 1 or m− l− n ≤ 0 when p = 1,then for 0 ≤ i ≤ m and v ∈Wm

p (T ), we have

||v − Ihv||W ip(T ) ≤ C(diam T )m−i|v|Wm

p (T ) (4.1.4)

where C depends on m, n, and T .

Theorem 4.1.2. Let all element sets of shape functions contain all polynomials ofdegree less than m and Th is non-degenerate. For v ∈ W k

p (Ω), and 0 ≤ s ≤ k ≤ m,we get

||v − Ihv||W sp (T ) ≤ Chk−sT |v|Wk

p (ST ), (4.1.5)

where the patch ST is a domain including the elements in Th neighboring the meshelement T , in other words

ST = interior(∪Ti | Ti ∩ T 6= 0, Ti ∈ Th

).

24

4.1. PREPARATION FOR ERROR ANALYSIS

In a curved domain Γ, we note that Γ is only approximated by a(Γh), not equal,and that all functions being approximated are defined on a(Γh). Due to this fact,these theorems will be written for a lift function vl to bound a posteriori errorestimate according to our assumptions.

The interpolants in the theorems require a certain amount of smoothness onthe part of the function being approximated. We use local Scott-Zhang type in-terpolation, so the smoothness of averaging over the mesh elements suffices. Thefunction v, that is being interpolated in our work, is not smooth on a(Ωh). Sincethe Scott-Zhang type interpolation is optimal on the patch due to its definition, inerror analysis we use Theorem 4.1.2.

Here, we also recall Trace theorem from [5], that will be used in the subsequentsection.

Theorem 4.1.3. (Trace theorem) Let a domain Ω have a Lipschitz boundary, andthat 1 ≤ p ≤ ∞. Then, there is a constant C such that

||v||Lp(∂Ω) ≤ C||v||1−1/pLp(Ω) ||v||

1/pW 1

p (Ω) ∀v ∈W 1p (Ω). (4.1.6)

The Trace theorem will be used on mesh elements T ∈ Th to prove the a posteriorierror estimate.

4.1.3 Relation between Sobolev SpacesDue to the domain Γ being approximated by a(Γh), Sobolev spaces, on the con-

tinuous surface a(Γh) and the discrete surface Γh, will appear in the error analysis.Here, we provide a relation between those spaces to define the error indicator foreach mesh element T .

First, we present the relation in the L2 norm and H1 semi-norm in the followinglemma, and then the relation in the H2 semi-norm is reviewed, which is derived in[8, p.24].

Lemma 4.1.4. Let T ∈ Th and v ∈ H1(a(T )) and v` ∈ H1(T ). Then, we have

||v`||L2(T ) ≤∣∣∣∣∣∣∣∣ 1µh

∣∣∣∣∣∣∣∣1/2L∞(T )

||v||L2(a(T )) (4.1.7)

and

|v`|H1(T ) ≤∣∣∣∣∣∣∣∣ 1µh

∣∣∣∣∣∣∣∣1/2L∞(T )

||Ph(I − dH)||L∞(a(T ))|v|H1(a(T )) (4.1.8)

Proof. By using µh(x)dσh(x) = dσ(a(x)), (4.1.7) can be obtained easily,

||v`||2L2(T ) =∫T

[v`(x)]2dσh =∫T

1µh

[v(a(x))]2µhdσh

≤∣∣∣∣∣∣∣∣ 1µh

∣∣∣∣∣∣∣∣L∞(T )

∫a(T )

[v(a(x))]2dσ

=∣∣∣∣∣∣∣∣ 1µh

∣∣∣∣∣∣∣∣L∞(T )

||v||2L2(a(T )).

25


The equation (2.2.15) in [10],

∇Γhv`(x) = Ph(I − dH)∇Γv(a(x)),

gives us (4.1.8) as follows:

|v`|2H1(T ) =∫T

[∇Γhv`]2dσh =

∫T

1µh

[∇Γhv`]2µhdσh

=∫T

1µh

[Ph(I − dH)∇Γv(a(x))]2µhdσh

≤∣∣∣∣∣∣∣∣ 1µh

∣∣∣∣∣∣∣∣L∞(T )

∫a(T )

[Ph(I − dH)∇Γv(a(x))]2dσ

≤∣∣∣∣∣∣∣∣ 1µh

∣∣∣∣∣∣∣∣L∞(T )

||Ph(I − dH)||2L∞(a(T ))

∫a(T )

[∇Γv(a(x))]2dσ

≤∣∣∣∣∣∣∣∣ 1µh

∣∣∣∣∣∣∣∣L∞(T )

||Ph(I − dH)||2L∞(a(T ))|v|2H1(a(T )).

Lemma 4.1.5. Let T ∈ Th and v ∈W 2p (a(T )) for some 1 ≤ p ≤ ∞. Then,

|v`|W 2p (T ) ≤

∣∣∣∣∣∣∣∣ 1µh

∣∣∣∣∣∣∣∣1/pL∞(T )

(||Ph(I − dH)||L∞(a(T ))|v|W 2

p (a(T ))

+[||PhH||L∞(a(T ))||n− (n · nh)nh||L∞(a(T ))

+ maxi=1,2,3

||dPhHxi ||L∞(a(T ))

]|v|W 1

p (a(T ))

),

(4.1.9)

where Hxi denotes the derivative of H with respect to xi.

In an a posteriori error estimate, we want the relation between |v`|H2(ST ) and||v||H2(a(ST )) over the patch ST . It can be easily shown that ||v||H2(T ) is greaterthan |v|H1(T ) and |v|H2(T ) in the following way

|v|H1(T ) ≤(||v||2L2(T ) + |v|2H1(T ) + |v|2H2(T )

)1/2= ||v||H2(T ), (4.1.10)

which is the same for |v|H2(T ), then by this relation we obtain

|v`|H2(ST ) ≤ αST||v||H2(a(ST )),

where

αST=∑T∈ST

∣∣∣∣∣∣∣∣ 1µh

∣∣∣∣∣∣∣∣1/2∞(T )

(||Ph(I − dH)||L∞(a(T ))

+ ||PhH||L∞(a(T ))||n− (n · nh)nh||L∞(a(T ))

+ maxi=1,2,3


).

26

4.2. ERROR ANALYSIS

4.2 Error AnalysisWe introduce the dual-weighted residual method, which we use to estimate the

L2-norm error, and then we present L2-norm and energy-norm error estimates withelement-wise error indicators.

4.2.1 Dual-weighted Residual MethodIn this part, we review goal-oriented a posteriori error estimation with respect

to certain target functionals or the so-called quantity of interest, where the error inthe quantity of interest is estimated by a sum of cell errors. The error is written asa product of the residual of the underlying primal problem and the correspondingadjoint or dual solution, see for example [6].

Let us review the following theorem from [5].

Theorem 4.2.1. (Riesz Representation Theorem) Any continuous linear functionalL on a Hilbert space H can be represented uniquely as

L(w) = (u,w) for some u ∈ H.

We introduce the method for the Laplace equation in (2.2.7), and its weakresidual with the exact solution u is

r(u, v) = a(u, v)− L(v) ∀ v ∈ V.

The basic idea in this method is to solve the dual problem with respect to somequantity of interest M ∈ L2(Ω)′. By Riesz representation theorem, it follows thatthere exists some φ ∈ L2(Ω) such that

M(w) = (φ,w) ∀ w ∈ L2(Ω).

Let us introduce the dual problem, −∆v = φ with boundary conditions, that allowsfor any u, w ∈ V ,

a(u,w) = a∗(w, u),where a∗ is the bilinear form of the dual problem. Then, we seek a finite elementsolution vh ∈ Sh such that

a∗(v, w) = (φ,w) = M(w) ∀ w ∈ Vh, (4.2.1)

and the error in the target functional gives

|M(e)| = a∗(v, e) = a(e, v) = r(uh, φ),

where uh is the discrete solution (2.2.7). In later steps, it can be observed that theerror can be bounded as follows:

|M(e)| = (R(uh), v)Γ =∑T∈Th

(R(uh), v)T ≤∑T∈Th

||R(uh)||L2(T )||v||L2(T ), (4.2.2)

where R(uh) is the strong residual with possible interior facet jump contributions,and v is the solution of the continuous dual problem. We note that for a curvedsurface, there is an additional geometric term in (4.2.1).

27


4.2.2 L2-error Estimate

Before we start the definition of the a posteriori error estimate, we summarizeour assumptions as follows: Γh is a polyhedral approximation to Γ, which is theunion of triangles whose corners are on Γ, and first order finite elements are used.

Let us recall the following fact from [10] in the following lemma.

Lemma 4.2.2. Let ψ ∈ H1(Γ), which has vanishing trace on the boundary ∂Γ andhas a vanishing mean value if ∂Γ = ∅. We get that, for ψh ∈ Sh,∫

Γ∇Γ(u− u`h)∇Γψdσ =

∫Γh

(f `µh + ∆Γhuh)(ψ` − ψh)dσh

− 12∑T∈Th

∫∂T

[[∇Γhuh]](ψ` − ψh)ds

−∫

ΓP [I −Ah]∇Γu

`h∇Γψdσ

+∫

Γh

(f `µh − fh)ψhdσh

≡ A+B + C +D,

(4.2.3)

where [[∇Γhuh]] is a jump term in the normal derivative across edges, which is

defined as [[∇Γhuh]] = ∇Γh

uh|T1 · ~n1 +∇Γhuh|T2 · ~n2.

In a posteriori error analysis of higher dimensions, a jump or a difference termappears. This term can be clarified as follows: let T1 and T2 be two neighboringtriangles with a common internal edge e. Also, let ~n1 and ~n2 be normals which liesin the plane of T1 and T2, respectively across the same edge; however, ~n1 6= − ~n2generally. So, the difference ∇Γh

uh|T1 · ~n1 +∇Γhuh|T2 · ~n2 is obtained when the term

∇Γhuh · ~n is computed on the boundary of each triangle. As there is a vanishing

trace if ∂Γ 6= ∅, [[∇Γhuh]] = 0 for the outer edges e on ∂Γ.

In our analysis, we assume that fh = µhf`, such that the last term D in Lemma

4.2.2 does not appear. Also, we let the function ψh be a Scott-Zhang type interpolantof ψ` denoted by ψh = Ihψ`.

Theorem 4.2.3. Let u and u`h be the continuous and discrete solutions to theproblem (3.2.1), respectively. Also, let M(w) = (w, φ) be a quantity of interest withweight φ ∈ L2(Γ), and let v ∈ H2(Γ) solve the dual problem

−∆Γv = φ on Γ (4.2.4)

with satisfactory conditions according to presence of ∂Γ. Then, for e = u− u`h,

|M(e)| <∼

∑T∈Th

ξ2T

1/2

S, (4.2.5)

28

4.2. ERROR ANALYSIS

withS = max

φ∈L2(Γ)||v||H2(a(Γ)), (4.2.6)

where, for a convex domain Γ, S ≤ ||φ||L2(Γ), and

ξT :=(h2TαST

||fh + ∆Γhuh||L2(T ) + h

3/2T αST

∣∣∣∣[[∇Γhuh]]

∣∣∣∣L2(∂T ) + ||βh∇Γh

uh||L2(T )),

(4.2.7)and

αST=∑T∈ST

∣∣∣∣∣∣∣∣ 1µh

∣∣∣∣∣∣∣∣1/2∞(T )

(||Ph(I − dH)||L∞(a(T ))

+ ||PhH||L∞(a(T ))||n− (n · nh)nh||L∞(a(T ))

+ maxi=1,2,3


),

βh = √µhP (I −Ah)(I − dH)−1(I − nThn

nh · n

).

(4.2.8)

Proof. The dual problem is used in the dual-weighted functional alongside withLemma (4.2.2) for fh = µhf

` and ψh = Ihψ`, then we obtain

|M(u)−M(u`h)| = (u− u`h, φ) = (u− u`h,−∆Γv)= (∇Γ(u− u`h),∇Γv)

=∫

Γh

(f `µh + ∆Γhuh)(v` − Ihv`)dσh

− 12∑T∈Th

∫∂T

[[∇Γhuh]](v` − Ihv`)ds

−∫

ΓP [I −Ah]∇Γu

`h∇Γvdσ

= A+B + C

We apply Hölder’s inequality and Theorem 4.1.2, then we get

A =∫

Γh

(fh + ∆Γhuh)(v` − Ihv`)dσh

≤∑T∈Th

||fh + ∆Γhuh||L2(T )||v` − Ihv`||L2(T )

≤∑T∈Th

CIh2T ||fh + ∆Γh

uh||L2(T )|v`|H2(ST )

≤∑T∈Th

CIh2TαST

||fh + ∆Γhuh||L2(T )||v||H2(a(ST ))

≤ CI

∑T∈Th

h4Tα

2ST||fh + ∆Γh

uh||2L2(T )

1/2∑T∈Th

||v||2H2(a(ST ))

1/2

29


Then, a finite union of patches ST gives a(Γh), and by a(Γh) being an approximationto Γ and the regularity of the adjoint problem, we get

A <∼

∑T∈Th

h4Tα

2ST||fh + ∆Γh

uh||2L2(T )

1/2

||φ||L2(Γ). (4.2.9)

With the use of Theorem 4.1.2 and Trace Theorem 4.1.3, the bound of B canbe written as follows:

||v` − Ihv`||L2(∂T ) ≤ ||v` − Ihv`||1/2L2(T )||v

` − Ihv`||1/2H1(T )

≤ CI(h2T |v`|H2(ST )

)1/2 (hT |v`|H2(ST )

)1/2

≤ CIh3/2T αST

||v||H2(a(ST ))

Then by using the last relation,

B =∑T∈Th

∫∂T

[[∇Γhuh]](v` − Ihv`)ds

≤∑T∈Th

∣∣∣∣[[∇Γhuh]]

∣∣∣∣L2(∂T )||v

` − Ihv`||L2(∂T )

≤∑T∈Th

CIh3/2T αST

∣∣∣∣[[∇Γhuh]]

∣∣∣∣L2(∂T )||v||H2(a(ST ))

≤ CI

∑T∈Th

h3Tα

2ST

∣∣∣∣[[∇Γhuh]]

∣∣∣∣2L2(∂T )

1/2∑T∈Th

||v||2H2(a(ST ))

1/2

The second term is bounded in the same manner as in A, then

B <∼

∑T∈Th

h3Tα

2ST

∣∣∣∣[[∇Γhuh]]

∣∣∣∣2L2(∂T )

1/2

||φ||L2(Γ). (4.2.10)

To bound the term C, we use the relation between ∇Γv`h(a(x)) and ∇Γh

vh(x)given in Lemma 3.1.1.

||P (I −Ah)∇Γu`h||2L2(a(T )) =

∫a(T )

[P (I −Ah)∇Γu

`h(a(x))

]2dσ(a(x))

=∫T

[√µhP (I −Ah)(I − dH)−1

(I − nThn

nh · n

)∇Γh

vh(x)]2

dσh

= ||βh∇Γhvh||2L2(T ),

whereβh = √µhP (I −Ah)(I − dH)−1

(I − nThn

nh · n

).

30

4.2. ERROR ANALYSIS

Then, the bound of C is as follows:

C =∫

ΓP (I −Ah)∇Γu

`h∇Γvdσ

≤ ||P (I −Ah)∇Γu`h||L2(Γ)||∇Γv||L2(Γ)

≈∑T∈Th

||P (I −Ah)∇Γu`h||L2(a(T ))|v|H1(a(T ))

=∑T∈Th

||βh∇Γhuh||L2(T )|v|H1(a(T ))

≤

∑T∈Th

||βh∇Γhuh||2L2(T )

1/2∑T∈Th

|v|H1(a(T ))

1/2

≤

∑T∈Th


1/2

||v||H2(a(Γ)),

since |v|H1(a(T )) ≤ ||v||H2(a(Γ)), and by the adjoint problem,

C <∼

∑T∈Th


1/2

||φ||L2(Γ). (4.2.11)

Hence, we combine (4.2.9), (4.2.10) and (4.2.11), we obtain

|M(e)| <∼

∑T∈Th

h4Tα

2ST||fh + ∆Γh

uh||2L2(T )

1/2

||φ||L2(Γ)

+

∑T∈Th

h3Tα

2ST

∣∣∣∣[[∇Γhuh]]

∣∣∣∣2L2(∂T )

1/2

||φ||L2(Γ)

+

∑T∈Th


1/2

||φ||L2(Γ),

(4.2.12)

and by using the inequality (k∑i=1

ai

)2

≤ k(

k∑i=1

a2i

)for k = 3 then, it follows that

|M(e)|2 <∼

∑T∈Th

h4Tα

2ST||fh + ∆Γh

uh||2L2(T ) +∑T∈Th

h3Tα

2ST

∣∣∣∣[[∇Γhuh]]

∣∣∣∣2L2(∂T )

+∑T∈Th


||φ||2L2(Γ).

31


By using the relation ||.||`32 ≤ ||.||`31 , we can write

∑T∈Th

(a2

1(T ) + a22(T ) + a2

3(T ))1/2

≤

∑T∈Th

(a1(T ) + a2(T ) + a3(T ))2

1/2

,

the bound for M(e) becomes

|M(e)| <∼

∑T∈Th

(h2TαST

||fh + ∆Γhuh||L2(T ) + h

3/2T αST

∣∣∣∣[[∇Γhuh]]

∣∣∣∣L2(∂T )

+||βh∇Γhuh||L2(T )

)2)1/2

||φ||L2(Γ),

and we define ξT , which is the error indicator in each cell T ∈ Th, as follows:

ξT := h2TαST

||fh + ∆Γhuh||L2(T ) + h

3/2T αST

∣∣∣∣[[∇Γhuh]]

∣∣∣∣L2(∂T ) + ||βh∇Γh

uh||L2(T )

The L2 error can be derived from choosing φ = u− u`h, and more generally theH−1-norm of the error e defined by Definition 2.1.9 can be bounded by the followingcorollary:

Corollary 4.2.4. Under the same conditions of Theorem 4.2.3 with the assumptionthat φ ∈ H1(Γ), we obtain

||e||H−1(Γ) <∼

∑T∈T〈

(h2TαST

||R(uh)||L2(T ) + h3/2T αST

∣∣∣∣[[∇Γhuh]]

∣∣∣∣2L2(∂T )

+||βh∇Γhuh(x)||L2(T )

)2)1/2

.

If we choose Γ to be a flat domain, then the error indicator can be found asfollows: the signed distance function d(x) = 0 for each x ∈ Γh since each point xstays on Γ, so, P = Ph = I, H = 0, and µh = 1 as a(x) = x. Hence, αST

= 1, andβh cannot be defined since n = 0, so we check the C term before using any bound:I − Ah = 0, which means there is no geometric component. Then, we define theerror indicator on a flat domain as follows:

ξT :=(h2T ||fh + ∆Γh

uh||L2(T ) + h3/2T

∣∣∣∣[[∇Γhuh]]

∣∣∣∣L2(∂T )

). (4.2.13)

Note that the strong residual and the jump terms are the standard terms on aflat domain; however, on embedded surfaces, they have also geometric information

32

4.2. ERROR ANALYSIS

due to projection of Sobolev spaces. The last term in (4.2.7) is the geometric term,and

||βh||L2(T ) ≤ ||P (I −Ah)||L2(Γ) ≤ Ch2T

see in [11]. The geometric term and the residual term including the jump is of thesame order. A generalized error bound including the data approximation term anddifferent approximation function to f l can be found in [8] and [12].

4.2.3 Energy Estimate

After some modification of Lemma 4.2.3, the error bound in energy-norm canbe found easily. We start from Lemma 4.2.2 for ψ = e = u− u`h ∈ H1(Γ), then wedefine

e`(x) = (u− u`h)(a(x)) for x ∈ Γh,

and eh = Ihe`. By Theorem 4.1.2 and equation (4.1.8) in the bound of the term A,we get

||e` − Ihe`||L2(T ) ≤ CIhT |v`|H1(ST ) ≤ CIhTγST|e|H1(a(ST )),

where

γST=∑T∈ST

∣∣∣∣∣∣∣∣ 1µh

∣∣∣∣∣∣∣∣1/2L∞(T )

||Ph(I − dH)||L∞(a(T )),

and by the Trace Theorem 4.1.3 in the bound of the term B,

||e` − Ihe`||L2(∂T ) ≤ ||e`||1/2L2(ST )||e

`||1/2H1(ST )

≤ CI(hT |e`|H1(ST )

)1/2 (|e`|H1(ST )

)1/2

≤ CIh1/2T γST

|e|H1(a(ST )).

For the C term, there is no change in |e|H1(a(T )), and so by combining the newbounds of the A, B and C terms and dividing both sides with ||∇e||L2(Γ), we canpresent the following corollary:

Corollary 4.2.5. Let u and u`h be the continuous and discrete solutions to theproblem (3.2.1), respectively. Then, for e = u− u`h we obtain

||∇Γe||L2(Γ) <∼

∑T∈Th

ξ2T

1/2

, (4.2.14)

where

ξT := hTγST||fh + ∆Γh

uh||L2(T ) + h1/2T γST

∣∣∣∣[[∇Γhuh]]

∣∣∣∣L2(∂T ) + ||βh∇Γh

uh||L2(T ),

(4.2.15)

33


and

γST=∑T∈ST

∣∣∣∣∣∣∣∣ 1µh

∣∣∣∣∣∣∣∣1/2L∞(T )

||Ph(I − dH)||L∞(a(T )), (4.2.16)

βh = √µhP (I −Ah)(I − dH)−1(I − nThn

nh · n

). (4.2.17)

We note that the geometric components ||βh∇Γhuh|| in L2-norm and energy-

norm error bounds are the same; however, in the energy-norm, the geometric compo-nent ||βh∇Γh

uh||L2(T ) decreases asymptotically, which can be observed in numericalimplementation part.

34

Chapter 5

Adaptive SFEM

In this chapter, the a posteriori error estimates, that is derived in the previoussection, will be used in adaptivity. First, we provide the Adaptive SFEM algorithm,that explains how the implementation part is established, and then numerical im-plementation results for a sphere and a torus are presented.

5.1 Numerical ImplementaionIn the numerical implementation part, our model has some simplifications de-

pending on the tests in [8] and [12]. In adaptivity, difference between mesh elementdiameters are high; however, if the mesh is sufficiently fine, we compare the resultswith the number of degrees of freedoms (DOF ) assuming

O(h2T ) ∼ 1

DOF, (5.1.1)

and the jacobian µh is approximately equal to 1. The order of αSTin (4.2.8) is

calculated as|1− αST

| = O(hT ),

and due to the performance of the adaptive algorithm, it is tested that one can setαST

= 1 without any negative effect on the accuracy of the L2-norm error bound.The term ∣∣∣1− ||Ph(I − dH)||L∞(a(T ))

∣∣∣ = O(h2T ) (5.1.2)

is dominated in αST, and so it is expected to decrease with order DOF−1 first,

then there is an increment to order DOF−1. We have the expression ||Ph(I −dH)||L∞(a(T )) in the term γST

(4.2.16), that is obtained in the energy-norm estimate.Due to (5.1.2), it would thus reasonable and efficient to take γST

is equal to 1. Aslast, we recall the geometric term bound:

||βh∇Γhuh||L2(T ) ≤ Ch2

T ||∇Γhuh||L2(T ), (5.1.3)

depending on the theoretical and test results in [11] and [12], respectively.

35

CHAPTER 5. ADAPTIVE SFEM

In this part, the meshes that we use are created in FEniCS [1] and SALOME [2],and we implement Adaptive SFEM for the Laplace-Beltrami problem in FEniCS. Westart by solving the Laplace-Beltrami equation with our iterative adaptive algorithmwhich can be described as follows:

i) A reference solution u to the Laplace-Beltrami equation is found on a fine mesh.

ii) Starting with a coarse mesh, we solve the primal problem

(∇Γhuh,∇Γh

wh) = (fh, wh) ∀ wh ∈ Vh,

and get a approximate solution uh to the problem.

iii) Then, the dual problem is solved according to the dual-weighted residual method,i.e.,

(∇Γhvh,∇Γh

wh) = M(wh) ∀ wh ∈ Vh,

see in (4.2.1). The goal of interest M(wh) has a goal marker to achieve adap-tivity in a specific region of the domain Γ, where the exact solution to theequation has peak values or singularities, and it has value 1 inside this regionand value 0 elsewhere.

iv) The error indicator is computed from uh and vh, which are the primal anddual solutions, respectively. In the do-nothing method, which is developed as ageneral method for error control of any PDE, the error indicator is defined as

ηT = r(uh, vh)T = a(uh, vh)− L(vh) = (∇Γhuh,∇Γh

vh)− (fh, vh),

where r is the weak residual, see for example [16]. The error indicator ηT is usedto decide which element to be refined; however, a posteriori error estimates arecalculated from the error indicator ξ, which are found in the theoretical part,see the error indicators ξT of the L2-norm error bound in (4.2.7), and of theenergy-norm error bound in (4.2.15). By that, we also see that the do-nothingmethod efficiently works as an adaptive method.

v) To derive the L2-norm and energy-norm errors from the difference of an approx-imate and reference solutions, u−uh, the reference solution and the approximatesolution must be on the same mesh. The solution on a coarse mesh is projectedon the fine mesh, where the errors are computed, not to loose information inthe reference solution.

vi) The L2-norm and energy-norm errors are also obtained by using the goal ofinterest. The L2-norm error is∣∣M(u)−M(uh)

∣∣ = ρ (u− uh, u− uh)

where ρ is a goal marker, and the energy-norm error is∣∣M(u)−M(uh)∣∣ = ρ (∇Γh

(u− uh),∇Γh(u− uh)) .

36

5.2. IMPLEMENTATION RESULTS

In this approach, we want to compute errors in regions that the goal of interestfocuses on.

vii) The residual, jump and geometric terms are found after the simplifications areapplied, then the theoretical error indicator ξT in each T ∈ Th is computed forboth L2-norm and energy-norm error bounds, and after summation over eachT , we obtain the desired a posteriori error estimates.

viii) In this step, we check the stopping criterion on the a posteriori error estimate,for example, the L2 error bound, i.e.,

E(uh, h, f) ≤ TOL,

which is described in Section 2.4. If the a posteriori error bound is less thanTOL, the adaptive algorithm is terminated, otherwise the algorithm continueswith the next step.

ix) We use ηT to select the local refinement in the current mesh. To choose whichmesh element T to be refined, we use a threshold 0.25, and we mark the cellT if its error indicator ηT is greater than the product of the threshold and themaximum error indicator in Th, i.e.,

ηT ≥ 0.25 maxT∈ThηT . (5.1.4)

x) After detecting which T to be refined, the mesh is refined, and due to workingon a curved surface, new nodes will not be replaced on the analytical surface Γ,instead those nodes are moved to the closest point on Γ by using the parametricrepresentations, see in (3.1.8) and (3.1.10). Aspect ratio is the ratio of theelement diameter and the diameter of the largest ball that can be contained inT , which is a measure of a mesh being shape regular, and the aspect ratios ofelements are checked and if needed, the nodes are moved to the midpoints oftheir neighboring cells, which is known as Lagrangian mesh smoothing.

5.2 Implementation ResultsThe given algorithm is applied to two different problems: a sphere and a torus,

and the results are presented in the following sections. The test cases are used ascomputational examples in [8] and [10].

5.2.1 Implementation on a Sphere

Let Γ be a unit sphere with spherical coordinates (θ, φ), introduced in (3.1.8).For 0 ≤ φ ≤ 5π/3, we set

u(φ, θ) = (sinθ)λsin(λφ), (5.2.1)

37


101 102 103 104 105 106

DOF

10-4

10-3

10-2

10-1

100

101

102

error

Uniform refinementL 2 errorE error

L 2 error(goal)

E error(goal)

L 2 boundE bound

DOF−1

DOF−. 5

(a)

101 102 103 104 105 106

DOF

10-5

10-4

10-3

10-2

10-1

100

101

102

err

or

Uniform refinement

Residual term(L 2bound)Residual term (E bound)

Jump term (L 2bound)

Jump term (E bound)

Geometric term

DOF−1

DOF−. 5

(b)

Figure 5.1: Results for uniform SFEM solving the Laplace-Beltrami equation,−∆Γu = f , on the unit sphere Γ with boundary: a) L2-norm and energy-normerrors, and their theoretical bounds with respect to DOF . b) Residual term includ-ing jump term, and geometric term of the a posterirori error estimates with respectto DOF .

with λ = 0.6. Then, u solves −∆Γu = f on Γ and has a homogeneous Dirichletboundary condition on ∂Γ with

f = λ(λ+ 1)(sinθ)λsinλθ, (5.2.2)

where the computation of f is provided in the Appendices.We note that the L2 error bound depends on the H2-regularity and that this

regularity only holds on convex polygonal domains, that is, every line segment thatconnects every pair of points within the domain lies in the domain [14]. In ourfirst test case, the unit sphere with boundaries is non-convex, therefore we do notexpect reliable L2 bound results; however, the energy-norm estimate is satisfied onnon-convex domains due to not using the convexity in the proof of Corollary 4.2.5.

Figure 5.2: A refined mesh obtained by Adaptive SFEM based on error indicatorηT on the unit sphere, and its magnified view along z-axis.

38


0.24999

0.49998

0.74997

-2e-15

1e+00

u

0.24999

0.49998

0.74997

-2e-15

1e+00

u

Figure 5.3: Adaptive SFEM solutions solving the Laplace-Beltrami equation on theunit sphere at the first and the last steps of the adaptive algorithm.

In Figure 5.1 a., the L2 error decreases slower than the L2 bound while theerror and the bound in the energy-norm have the same order of convergence. Thisconfirms that the theoretical error bound does not provide accurate results on non-convex domains. Also, the energy-norm error and energy-norm goal error, that isE error(goal) and derived from the goal of interest, are almost the same while forL2 error, L2 goal error is lower than the other. The right figure shows that thejump error components become dominant in both bounds asymptotically, and thegeometric term decreases gradually and looses its importance in the energy-norm.

We run the same example with the adaptive algorithm starting with an initialmesh of 12 elements. Since u has singularity at the poles, we get heavy refinement onthese regions, which can be seen in Figure 5.2. The presence of a singularity causesa high residual error term including jump at the poles, therefore we expect thatrefined elements in these regions have high residual and jump terms in comparisonwith the geometric component.

Figure 5.4: Refined cells with respect to the L2 error bound at the second and thelast steps of the adaptive algorithm. Blue cells are not refined, and red and greenones indicate the refined cells where the residual term including jump over facets,and the geometric terms are dominant, respectively.

39


101 102 103 104 105 106

DOF

10-5

10-4

10-3

10-2

10-1

100

101

102

Error

Adaptive refinement

L 2 errorE error

L 2 error(goal)

E error(goal)

L 2 boundE bound

DOF−1

DOF−. 5

(a)

101 102 103 104 105 106

DOF

10-5

10-4

10-3

10-2

10-1

100

101

102

Err

or

Adaptive refinement

Residual+jump term (L 2bound)

Residual+jump term (E bound)

Geometric term

DOF−1

DOF−. 5

(b)

Figure 5.5: Results for Adaptive SFEM on the unit sphere Γ based on error indicatorηT : a) L2-norm and energy-norm errors, and their bounds with respect to DOF .b) Residual component including the jump, and geometric components of the errorswith respect to DOF .

The Galerkin solution uh is plotted in Figure 5.3 for the first and the last steps ofthe adaptive algorithm. This figure shows that the function u has the peak value 1 atthe back of the sphere, so Figure 5.4 confirms that refined elements on the backsideof the sphere have greater residual and jump terms while on the region closer tothe boundary, elements are refined where the geometric component is dominant. InFigure 5.4 and Figure 5.6, the error components of the L2 error bound are presenteddue to being more illustrative than those of the energy-norm error bound, whichthe total of the residual and the jump terms are dominates to the geometric termmore at each iteration, see Figure 5.5. The percentage of elements to be refinedwhere the residual and jump terms are dominant over the other term can be seenin Figure 5.6; the number of refined elements with high geometric error is less.

101 102 103 104 105 106

Degrees of freedoms

75

80

85

90

95

100

Percentage

Percentage of elements

Figure 5.6: Percentage of refined elements whose residual component including thejump term is greater than the geometric component.

40


The sphere test case in L2-norm is considered in [8], which presents two differentestimators: a L2 a posteriori error estimator and a pointwise error estimator, whichthe Green’s function is used as the auxiliary function and the work provides areliable estimator for non-convex polygonal domains.

In Adaptive SFEM, Figure 5.5, the L2 errors and its bound have the sameconvergence after some tolerance, which is similar to the energy-norm ones, althoughthe L2 error bound does not give an accurate upper bound for L2 error in uniformrefinement. It is also possible to observe the same relation between errors obtainedby u−uh and the goal of interest. We note that the slower convergence arises fromthe simplifications that we used on the geometric information, αST

and γST, and

the geometric error, ||βh∇Γhuh||; however, after some refinement steps, those terms

converges to their limit, so the required rate of convergence is observed.

5.2.2 Implementation on a Torus

101 102 103 104 105 106

DOF

10-3

10-2

10-1

100

101

102

Error

Uniform refinement

L 2 errorE error

L 2 error(goal)

E error(goal)

L 2 boundE bound

DOF−1

DOF−. 5

(a)

101 102 103 104 105 106

DOF

10-3

10-2

10-1

100

101

102

Err

or

Uniform refinement



Geometric term

DOF−1

DOF−. 5

(b)

Figure 5.7: Results for uniform SFEM solving the Laplace-Beltrami equation,−∆Γu = f , on a torus Γ with major and minor radii 1 and 0.25, and u =exp

(1

1.85−x2

)siny: a) L2-norm and energy-norm errors and their error bounds with

respect to DOF . b) Error bound components with respect to DOF .

In our second test case, we set Γ to be a torus, which is obtained by revolvingthe circle (x− 1)2 + z2 = (0.25)2 about z-axis, then the minor and major radii are0.25 and 1, respectively. As a test solution, we take the function

u = exp( 1

1.85− x2

)siny, (5.2.3)

which has exponential peaks at ±x-axes when value x reaches to ±1.25. The sourcefunction f is calculated from u by using the Laplacian operator on a Riemannmanifold, see the Appendices.

We start the experiments with uniform refinement that is presented in Figure5.7, and the errors and the bounds have optimal order of convergence on the torus.

41


Figure 5.8: Elements marked for refinement at an intermediate and the last steps ofthe adaptive algorithm based on the L2 error bound. Blue elements are preservedfor the next iteration, red and green elements are marked where the residual errorwith the jump term, and the geometric error are dominant, respectively.

In Adaptive SFEM, the test begins with 40 degrees of freedoms and goes up to325696. Elements only with high residual term are refined in the first six steps, andwe can observe refined elements with dominant geometric terms later, see Figure 5.8.The exponential peak values that lie on x-axes cause mainly to focus on elementswhere the geometric error is less important, so that, we see heavy refinement onthese regions, which is presented in Figure 5.10. The reference solution is obtainedon a mesh with 974791 degrees of freedoms owing to heavy computation; makingthe reference mesh finer may show better convergence rate for the L2 error andthe bound than we have obtained in Figure 5.9. However, we do not see slowerconvergence in L2 goal error in comparison with L2 error. The reason of betterconvergence in L2 goal error might be to compute it on the regions where the peakvalues of the function is located, and so, heavy refinement is required; hence, theelements diameters are closed to each other which satisfy the relation (5.1.1).

101 102 103 104 105 106

DOF

10-4

10-3

10-2

10-1

100

101

102

Error

Adaptive refinement

L 2 errorE error

L 2 error(goal)

E error(goal)

L 2 boundE bound

DOF−1

DOF−. 5

(a)

101 102 103 104 105 106

DOF

10-3

10-2

10-1

100

101

102

Err

or

Adaptive refinement



Geometric term

DOF−1

DOF−. 5

(b)

Figure 5.9: Results for Adaptive SFEM on the torus based on error indicator ηT :a) L2 and energy-norm errors with their bounds with respect to DOF . b) Errorbound components with respect to DOF .

42


-1.3913

0

1.3913

-3e+00

3e+00

u

-1.3913

0

1.3913

-3e+00

3e+00

u

Figure 5.10: Meshes and Adaptive SFEM solutions on the torus. Above: Meshes atthe first and an intermediate steps of the adaptive algorithm. Below: FEM solutionsat the first and the last steps of the adaptive algorithm.

43

Chapter 6

Conclusion and Future Work

In this part, we discuss our theoretical and implementation results, and howthese results can be improved in the future work.

6.1 Theoretical PartIn this report, the model Laplace-Beltrami equation is formulated on a domain

without boundary or with boundary satisfying a vanishing trace, and we have triedto find a posteriori error estimates for SFEM with first-order degree elements on atriangulated manifold. In the theoretical part, we have introduced some methods,theorems and known facts. After we set some assumptions, we have derived aposteriori error estimates, which is divided into a residual error term includingjump with explicit geometric information, and a geometric error term in Chapter 3following on [8] and [12].

In future work, constructing a lower bound to the a posteriori error estimateshelps in efficiency analysis, then the upper and the lower bounds can show theefficiency of the grid modification. The Laplace-Beltrami equation can be consideredwith non-homogeneous boundary conditions, which can lead additional terms in thebilinear form. In this case, the assumption, that we built in the Green’s theorem formanifolds in Section 2.2, would be efficient to use in theory and in implementationparts. Also, the order of FEM element can be increased which is introduced in [9].

6.2 Implementation PartIn the implementation part, we have made some simplifications of the a posteriori

error estimators and presented two test cases: the unit sphere with boundary anda torus as a non-convex and convex polygonal domains, respectively. The uniformrefinement results on the sphere have confirmed that the L2 bound does not holdfor non-convex polygonal domains, while we have observed the same convergence,DOF−1 in Adaptive SFEM; however, for the energy-norm bound, the error and theestimator have reached desired order of convergence in both cases. In the second

45

CHAPTER 6. CONCLUSION AND FUTURE WORK

test case, the optimal order of convergence is obtained for the L2-norm bound onthe torus, and in the adaptive algorithm, the results show that the estimators andthe errors are decreasing slightly slower than the expected rate while L2 goal errorconverges with the rate of DOF−1.

The implementation results can be improved in few steps in future work. Firststep would be to run the test cases in parallel, in particular for the torus, which isexpensive in computation. By avoiding the simplification, which means computingterms including geometric information and geometric error, we can aim to havemore accurate error bounds.

Here, we can also note that a manifold can be approximated by tetrahedronelements, which means a mesh is built in a tubular region of width ε, Uε, aroundthe surface. Then, for a sufficiently small ε value, we expect the Galerkin solutionto converge the solution which is obtained from a fine triangulated domain. Witha fine enough tetrahedral mesh, it might be possible to achieve better solution,especially for the functions which have singularities on manifold Γ.

46

Appendices

47

A.1. LAPLACIAN ON A SPHERE

A.1 Laplacian on a SphereFor a unit sphere, the exact solution is given by u(φ, θ) = (sinθ)λsin(λφ) and we

want to find the Laplacian of this function. First, we find required derivatives,∂u

∂φ= λ(sinθ)λcos(λφ),

∂2u

∂φ2 = −λ2(sinθ)λsin(λφ),

∂u

∂θ= λ(cosθ)λ−1cosθsin(λφ).

Then, the Laplacian can be written from (3.1.9) as follows:

∆Γu = 1sinθ

∂u

∂θ

(sinθ∂u

∂θ

)+ 1

sin2θλ2 ∂

2u

∂φ2

= 1sinθλsin(λφ) ∂

∂θ

((sinθ)λcosθ

)− 1

sin2θλ2(sinθ)λsin(λφ)

= 1sinθλsin(λφ)

(λ(sinθ)λ−1cos2θ + (sinθ)λsinθ

)− 1

sin2θλ2(sinθ)λsin(λφ)

=− λ2(sinθ)λ−2sin(λφ)(1− cos2θ)− λ(sinθ)λsin(λφ)=− λ(λ+ 1)(sinθ)λsin(λφ).

Thus, the source function is

f = λ(λ+ 1)(sinθ)λsin(λφ).

A.2 Laplacian on a Torus

The exact solution on a torus is given by u(x, y, z) = exp(

1k−x2

)siny, where k

is a scalar value. By using the parametric representation in (3.1.10), it is written as

u(φ, θ) = exp (w) sin(rsinφ),

wherew = 1

k − r2cos2φ and r = R+ rcosθ.

The following derivatives will be used in the computation of the Laplacian.

∂

∂θr = −rsinθ, ∂2

∂θ2 r = −rcosθ

∂

∂θw = −2rrsinθcos2φw2,

∂

∂φw = −2r2sinφcosφw2

The Laplacian on a torus in (3.1.11) can be written as

∆Γu = 1r2r

∂

∂θ

(r∂

∂θu

)+ 1r2

∂2

∂φ2u = I + II

49


To compute I, first we will compute them partially.

∂

∂θu = ∂

∂θ(ewsin(rsinφ))

= sin(rsinφ)ew ∂

∂θw + ∂

∂θ(rsinφ)cos(rsinφ)ew

= −2rrsinθcos2φsin(rsinφ)eww2 + rsinθsinφcos(rsinφ)ew

= −rsinθ(2rcos2φsin(rsinφ)eww2 + sinφcos(rsinφ)ew

)We apply chain rule to I, and the derivations which will be used are:

∂

∂θ(rrsinθ) = r(−rsin2θ + rcosθ), (A.2.1)

∂

∂θ

(2rcos2φsin(rsinφ)eww2

)= 2cos2φ

[∂

∂θ(r)sin(rsinφ)eww2 + r

∂

∂θ(sin(rsinφ)) eww2

+rsin(rsinφ) ∂∂θ

(ew)w2 + rsin(rsinφ)ew ∂

∂θ

(w2)]

= 2cos2φ[−rsinθsin(rsinφ)eww2 − rrsinθsinφcos(rsinφ)eww2

−2rr2sinθcos2φsin(rsinφ)eww4 − 4rr2sinθcos2φsin(rsinφ)eww3],

(A.2.2)

∂

∂θ(sinφcos(rsinφ)ew)

= sinφ[∂

∂θ(cos(rsinφ)) ew + cos(rsinφ) ∂

∂θ(ew)

]= sinφ

[rsinθsinφsin(rsinφ)ew − 2rrsinθcos2φcos(rsinφ)eww2

].

(A.2.3)

Hence, I will follow by (A.2.1), (A.2.2) and (A.2.3),

∂

∂θ

(r∂

∂θu

)= ∂

∂θ

(−rrsinθ

(2rcos2φsin(rsinφ)eww2 + sinφcos(rsinφ)ew

))= r(−rsin2θ + rcosθ)

[2rcos2φsin(rsinφ)eww2 + sinφcos(rsinφ)ew

]− rr2sinθ

[−2sinθcos2φsin(rsinφ)eww2

−2rsinθsinφcos2φcos(rsinφ)eww2

−4r2sinθcos4φsin(rsinφ)eww4 − 8r2sinθcos4φsin(rsinφ)eww3

+sinθsin2φsinφ)ew − 2rsinθsinφcos2φcos(rsinφ)eww2].

50

A.2. LAPLACIAN ON A TORUS

Then, I follows

1r2r

∂

∂θ

(r∂

∂θu

)= 2cos2φsin(rsinφ)eww2

(2sin2θ − r

rcosθ

)+ 4sin2θsinφcos2φcos(rsinφ)eww2

+ 4r2sin2θcos4φsin(rsinφ)eww3 (w + 2)− sin2θsin2φsin(rsinφ)ew

+ sinθcos(rsinφ)ew(1rsinθsinφ− 1

rcosθ

).

(A.2.4)

To compute II,

∂

∂φu = ∂

∂φ(ewsin(rsinφ))

= sin(rsinφ) ∂∂φ

(ew) + ew∂

∂φ(sin(rsinφ))

= −2r2sinφcosφsin(rsinφ)eww2 + rcosφcos(rsinφ)ew,

∂

∂φ

(−2r2sinφcosφsin(rsinφ)eww2

)= −2r2

[cos2φsin(rsinφ)eww2 − sin2φsin(rsinφ)eww2

+rsinφcos2φcos(rsinφ)eww2 − 2r2sin2φcos2φsin(rsinφ)eww4

−4r2sin2φcos2φsin(rsinφ)eww3],

(A.2.5)

∂

∂φ(rcosφcos(rsinφ)ew) = r

[sinφcos(rsinφ)ew − rcos2φsin(rsinφ)ew

−2r2sinφcos2φcos(rsinφ)eww2].

(A.2.6)

By (A.2.5) and (A.2.6), we get

1r2

∂2

∂φ2u = ∂

∂φ

(−2r2sinφcosφsin(rsinφ)eww2 + rcosφcos(rsinφ)ew

)= −2cos2φsin(rsinφ)eww2 + 2sin2φsin(rsinφ)eww2

− 2rsinφcos2φcos(rsinφ)eww2 + 4r2sin2φcos2φsin(rsinφ)eww4

+ 8r2sin2φcos2φsin(rsinφ)eww3 − 1rsinφcos(rsinφ)ew

− cos2φsin(rsinφ)ew − 2rsinφcos2φcos(rsinφ)eww2.

Then, II follows

51


1r2

∂2

∂φ2u = −2sin(rsinφ)eww2cos2(2φ)− 4rsinφcos2φcos(rsinφ)eww2

+ 4r2sin2φcos2φsin(rsinφ)eww3 (w + 2)− 1rsinφcos(rsinφ)ew

− cos2φsin(rsinφ)ew.

(A.2.7)

After combining I and II terms in (A.2.4) and (A.2.7), respectively, we obtain

∆Γu = 4sinφcos2φcos(rsinφ)w2ew(sinθ2 − r)+ 4r2cos2φsin(rsinφ)w3ew(w + 2)(sin2θcos2φ+ sin2φ)

+ 2sin(rsinφ)w2ew[cos2φ(2sin2θ − r

rcosθ)− cos(2θ)

]+ cos(rsinφ)ew

[sinθ

(1rsinθsinφ− 1

rcosθ

)− 1rsinφ

]− sin(rsinφ)ew(sin2θsin2φ− cos2φ).

(A.2.8)

Hence, the source function f = −∆Γu.

52

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