bb-sinum-1

8/14/2019 bb-sinum-1

1/28

NUMERICAL ANALYSIS OF A SECOND-ORDER PURE

LAGRANGE-GALERKIN METHOD FOR

CONVECTION-DIFFUSION PROBLEMS. PART I: TIME

DISCRETIZATION

MARTA BENITEZ AND ALFREDO BERMUDEZ

Abstract. We propose and analyze a second order pure Lagrangian method for variable co-

efficient convection-(possibly degenerate) diffusion equations with mixed Dirichlet-Robin boundaryconditions. First, the method is rigorously introduced for exact and approximate characteristics.Next, l(H1) stability is proved and l(H1) error estimates of order O(t2) are obtained. More-over, l(L2) stability and l(L2) error estimates of order O(t2) with constants bounded in thehyperbolic limit are shown. For the particular case of Dirichlet boundary conditions, diffusion tensorA = Iand right-hand side f = 0, the l(H1) stability estimate is independent of . In a secondpart of this work, the pure Lagrangian scheme will be combined with Galerkin discretization usingfinite elements spaces and numerical examples will be presented.

Key words. convection-diffusion equation, pure Lagrangian method, characteristics method,stability, error estimates, second order schemes

AMS subject classifications. 65M12, 65M15, 65M25, 65M60

1. Introduction. The main goal of the present paper is to introduce and ana-

lyze a second order pure Lagrangian method for the numerical solution of convection-diffusion problems with possibly degenerate diffusion. Computing the solutions ofthese problems, especially in the convection dominated case, is an important andchallenging problem that requires development of reliable and accurate numericalmethods.

Linear convection-diffusion equations model a variety of important problems fromdifferent fields of engineering and applied sciences, such as thermodynamics, fluid me-chanics, and finance (see for instance [20]). In many cases the diffusive term is muchsmaller than the convective one, giving rise to the so-called convection dominatedproblems (see [17]). Furthermore, in some cases the diffusive term becomes degener-ate, as in some financial models (see, for instance, [26]).

This paper concerns the numerical solution of convection-diffusion problems withdegenerate diffusion. For this kind of problems, methods of characteristics for time

discretization are extensively used (see the review paper [17]). These methods arebased on time discretization of the material time derivative and were introduced inthe beginning of the eighties of the last century combined with finite-differences orfinite elements for space discretization. When these methods are applied to the formu-lation of the problem in Lagrangian coordinates (respectively, Eulerian coordinates)they are called pure Lagrangian methods (respectively, semi-Lagrangian methods).The characteristics method has been mathematically analyzed and applied to differ-ent problems by several authors, primarily the semi-Lagrangian methods. In par-ticular, the (classical) semi-Lagrangian method is first order accurate in time. Ithas been applied to time dependent convection-diffusion equations combined with fi-

This work was supported by Xunta de Galicia under research project INCITE09 207 047 PR, andby Ministerio de Ciencia e Innovacion (Spain) under research projects Consolider MATHEMATICA

CSD2006-00032 and MTM2008-02483.Dep. de Matematica Aplicada, Universidade de Santiago de Compostela, Campus Vida, 15782Santiago de Compostela, Spain ([email protected], [email protected]). The first au-thor was supported by Ministerio de Educacion.

1

8/14/2019 bb-sinum-1

2/28

2 M. BENITEZ AND A. BERMUDEZ

nite elements ([16], [21]), finite differences ([16]), etc. Its adaptation to steady stateconvection-diffusion equations has been developed in [8] and, more recently, the com-bination of the classical first order scheme with disconuous Galerkin methods hasbeen used to solve first-order hyperbolic equations in [3], [2] and [4]. Higher or-der characteristics methods can be obtained by using higher order schemes for thediscretization of the material time derivative. In [22] multistep Lagrange-Galerkinmethods for convection-diffusion problems are analyzed. In [11] and [12] multistepmethods for approximating the material time derivative, combined with either mixed

finite element or spectral methods, are studied to solve incompressible Navier-Stokesequations. Stability is proved and optimal error estimates for the fully discretizedproblem are obtained. In [25] a second order characteristics method for solving con-stant coefficient convection-diffusion equations with Dirichlet boundary conditions isstudied. The Crank-Nicholson discretization has been used to approximate the ma-terial time derivative. For a divergence-free velocity field vanishing on the boundaryand a smooth enough solution, stability and error estimates are stated (see also [9]and [10] for further analysis). In [15] semi-Lagrangian and pure Lagrangian meth-ods are proposed and analyzed for convection-diffusion equation. Error estimates fora Galerkin discretization of a pure Lagrangian formulation and for a discontinuousGalerkin discretization of a semi-Lagrangian formulation are obtained. The estimatesare written in terms of the projections constructed in [13] and [14].In the present paper, a pure Lagrangian formulation is used for a more general prob-

lem. Specifically, we consider a (possibly degenerate) variable coefficient diffusive terminstead of the simpler Laplacian, general mixed Dirichlet-Robin boundary conditionsand a time dependent domain. Moreover, we analyze a scheme with approximatecharacteristic curves.

The mathematical formalism of continuum mechanics (see for instance [18]) isused to introduce the schemes and to analyze the error. In most cases the exactcharacteristics curves cannot be determined analytically, so our analysis include, as anovelty with respect to [15], the case where the characteristics curves are approximatedusing a second order Runge-Kutta scheme. A proof ofl(L2) stability inequality isdeveloped which can be appropriately used to obtain l(L2) error estimates of orderO(t2) between the solutions of the time semi-discretized problem and the contin-uous one; these estimates are uniform in the hyperbolic limit. More precisely, let

m ={nm}Nn=0 and m,t ={nm,t}Nn=0 denote, respectively, the exact solution ofthe continuous problem in Lagrangian coordinates (see3), and the discrete solutionof the pure Lagrangian method proposed and analyzed in this paper (see 4). Weprove (Corollary 4.12 and Theorem 4.27) the following inequalities:m,t

l(L2())+

4

BS[m,t]l2(L2())

+ S[m,t]

l2(L2(R))

J1

||0m,t||+ f XRK

l2(L2())+ g XRK

l2(L2(R))

,

(1.1)

and

|| m m,t||l(L2())+

4 BS[m m,t]

l2(L2())

+ S[m m,t]l2(L2(R))

J2 t2 ||m||C3(L2())(1.2)

+||m||C2(H1())+ ||m m||C2(L2(R))+ ||m||C2(L2(R))

8/14/2019 bb-sinum-1

3/28

HIGHER ORDER PURE LAGRANGIAN METHOD 3

+||fm||C2(L2())+ ||f||C1(T)+ ||gm||C2(L2(R))+ ||g||C1(TR

)

,

where m := Xe for a spatial field , being Xe the motion, B is the matrixB=

In1

, whereIn1 is then1 n1 identity matrix,S[] :={n+1 + n}N1n=0

for a sequence ={}Nn=0 and XRK ={XnRK}Nn=0 is a second order Runge-Kuttaapproximation ofXe. The diffusion tensor has the form A =

An1

and is

a uniform lower bound for the eigenvalues ofAn1 . Here, J1 does not depend on thediffusion tensor andJ2is bounded in the hyperbolic limit. Moreover, for the particularcase of Dirichlet boundary conditions, diffusion tensor A = B and right-hand sidef= 0, the l(H1) stability estimate is independent of (see Remark 4.9).

Similar stability and error estimates of order O(t2) are proved in the l(H1)norm. In particular, we prove (Corollary 4.16 and Theorem 4.28) the inequalities, Rt[m,t]

l2(L2())+

2

Bm,tl(L2())

+m,t

l(L2(R))

J3

2

B0m,t+ 0m,tR + f XRKl2(L2()) + g XRKl(L2(R)) + Rt[g XRK]l2(L2(R)) ,

(1.3)

and Rt[m m,t]l2(L2())

+

2

B (mm,t)l(L2())

+ m m,t

l(L2(R))J4 t2

||m||C3(L2())+ ||m||C2(H1())+||m m||C3(L2(R))+ ||m||C3(L2(R))+ ||fm||C2(L2())

+||f||C1(T)+ ||gm||C3(L2(R))+ ||g||C2(TR

)

,

(1.4)

where Rt[] :=

n+1 n

t N1

n=0

for a sequence

={}Nn=0. Here, J3 and J4

depend on the diffusion tensor; however for the particular case of diffusion tensor ofthe formA = B,J3 does not depend on it andJ4 is bounded in the hyperbolic limit.

To prove these estimates we assume that the exact solution and data of theproblem are smooth, and t is sufficiently small.

The paper is organized as follows. In Section 2 the convection-diffusion Cauchyproblem is stated in a time dependent bounded domain and notations concerningmotions and functional spaces are introduced. In Section 3, the strong formulationof the convection-diffusion Cauchy problem is written in Lagrangian coordinates andthe standard associated weak problem is obtained. In Section 4, a second ordertime discretization scheme is proposed for both exact and second order approximatecharacteristics. Next, under suitable hypotheses on the data, the l(L2) andl(H1)stability results are proved for small enough time step. Finally, assuming greaterregularity on the data, l(L2) and l(H1) error estimates of order O(t2) for the

solution of the time discretized problem are derived. In a second part of this work(see [7]), a fully discretized pure Lagrange-Galerkin scheme by using finite elementsin space will be analyzed and numerical results will be presented.

8/14/2019 bb-sinum-1

4/28


2. Statement of the problem and functional spaces. Let be a boundeddomain in Rd (d = 2, 3) with Lipschitz boundary divided into two parts: =D R, with D R = . LetTbe a positive constant andXe : [0, T] Rdbe a motion in the sense of Gurtin [18]. In particular,Xe C3( [0, T]) and foreach fixedt[0, T], Xe(, t) is a one-to-one function satisfying

det F(p, t)> 0 p,(2.1)

being F(, t) the Jacobian matrix of the deformation Xe(

, t). We call t = Xe(, t),

t = Xe(, t), Dt = Xe(D, t) y Rt = Xe(R, t), for t [0, T]. We assume that0 = . Let us introduce the trajectoryof the motion

T :={(x, t) :x t, t[0, T]},(2.2)

and the set

O :=

t[0,T]

t.(2.3)

For eacht, Xe(, t) is a one-to-one mapping from onto t; hence it has an inverse

P(, t) : t,(2.4)

such that

Xe(P(x, t), t) = x, P(Xe(p, t), t) = p (x, t) T (p, t) [0, T].(2.5)

The mapping

P :T ,

so defined is called the reference map of motion Xe and P C3(T) (see [18] pp.65 66). Let us recall that the spatial description of the velocity v :T Rd isdefined by

v(x, t) := Xe(P(x, t), t)

(x, t)

T.(2.6)

We denote byL the gradient ofv with respect to the space variables.Let us consider the following initial-boundary value problem.

(SP) STRONG PROBLEM. Find a function :T R such that

(x)

t(x, t) +(x)v(x, t) grad (x, t) div(A(x)grad (x, t)) = f(x, t),(2.7)

forx t and t (0, T), subject to the boundary conditions

(, t) =D(, t) onDt ,(2.8)(, t) +A()grad (, t) n(, t) = g(, t) onRt ,(2.9)

for t (0, T), and the initial condition(x, 0) =0(x) in.(2.10)

8/14/2019 bb-sinum-1

5/28


In the above equations, A :O Sym denotes the diffusion tensor field, whereSym is the space of symmetric tensors in the d-dimensional space, :O R,f :T R, 0 : R, D(, t) : Dt R and g (, t) : Rt R, t(0, T), aregiven scalar functions, andn(, t) is the outward unit normal vector to t.In the followingAdenotes a bounded domain in Rd. Let us introduce the Lebesquespaces Lr(A) and the Sobolev spaces Wm,r(A) with the usual norms|| ||r,A and| | | |m,r,A, respectively, for r= 1, 2, . . . , and m an integer. For the particular caser = 2, we endow space L2(A) with the usual inner product, A, which induces anorm to be denoted by|| ||A (see [1] for details). Moreover, we denote by H

1

D()the closed subspace ofH1() defined by

H1D() :=

H1(), |D0

.(2.11)

For a Banach function spaceXand an integer m, space Cm([0, T], X) will be abbre-viated as Cm(X) and endowed with norm

||||Cm(X) := maxt[0,T]

max

j=0,...,m||(j)(t)||X

.

In the above definitions, (j) denotes the j -th derivative of with respect to time.Finally, vector-valued function spaces will be distinguished by bold fonts, namelyLr

(A), Wm,r

(A) andHm

(A), and tensor-valued function spaces will be denoted byLr(A), Wm,r(A) and Hm(A). For the particular casem = 1 andr =, we considerthe vector-valued spaceW1,(A) equipped with the following equivalent norm to theusual one

||w||1,,A:= max {||w||,A, || div w||,A, ||w||,A} ,(2.12)

being

||w||,A := ess supxA

||w(x)||2,(2.13)

where|| ||2 denotes the tensor norm subordinate to the euclidean norm in Rd.Remark 2.1. For the sake of clarity in the notation, in expressions involving

gradients and time derivatives we use the following notation (see, for instance, [18]):1. We denote byp the material points in , and by x the spatial points in t

with t >0.2. A material field is a mapping with domain [0, T] and a spatial field is a

mapping with domainT.3. We define the material description m of a spatial field by

m(p, t) = (Xe(p, t), t).(2.14)

Similar definition is used for functions, , defined in a subset ofT or ofO.4. If is a smooth material field, we denote by(respectively, by Div ) the

gradient (respectively, the divergence) with respect to the first argument, and by the partial derivative with respect to the second argument (time).

5. If is a smooth spatial field, we denote by grad (respectively, div ) thegradient (respectively, the divergence) with respect to the first argument, and by

the partial derivative with respect to the second argument (time).

8/14/2019 bb-sinum-1

6/28


3. Weak formulation. We are going to develop some formal computations inorder to write a weak formulation of problem (SP) in Lagrangian coordinates p. First,by using the chain rule, we have

m(p, t) = (Xe(p, t), t) + grad (Xe(p, t), t) v(Xe(p, t), t).(3.1)

Next, by evaluating equation (2.7) at point x = Xe(p, t) and then using (3.1), weobtain

m(p, t)m(p, t) [div(A grad )]m(p, t) = fm(p, t),(3.2)for (p, t) (0, T). Note that in (3.2) there are derivatives with respect to theEulerian variable x. In order to obtain a strong formulation of problem (SP) inLagrangian coordinates we introduce the change of variable x = Xe(p, t). By usingthe chain rule we get (see [6])

[div(A grad )]m= Div

F1AmFTmdet F

1det F

.

Then, m satisfies

mmdet F Div

F1AmF

Tmdet F

=fmdet F.(3.3)

Throughout this article, we use the notationAm(p, t) := F1(p, t)Am(p, t)FT(p, t)det F(p, t) (p, t) [0, T],m(p, t) :=|FT(p, t)m(p)| det F(p, t) (p, t) [0, T],where m is the outward unit normal vector to . By using the chain rule and notingthat

n(Xe(p, t), t) = FT(p, t)m(p)

|FT(p, t)m(p)| (p, t) (0, T),

we get

A(x)grad (x, t) n(x, t) =F1(p, t)Am(p, t)FT(p, t)m(p, t) m(p)

|FT(p, t)m(p)| ,

for (p, t) (0, T) andx = Xe(p, t). Thus, from (2.8)-(2.10) and (3.3), we deducethe following pure Lagrangian formulation of the initial-boundary value problem (SP):

(LSP) LAGRANGIAN STRONG PROBLEM. Find a function m : [0, T] R such that

m(p, t)m(p, t)det F(p, t) Div Am(p, t)m(p, t) =fm(p, t)det F(p, t),(3.4)

for (p, t) (0, T), subject to the boundary conditions

m(p, t) = D(Xe(p, t), t) on

D

(0, T),(3.5)m(p, t)m(p, t) +Am(p, t)m(p, t) m(p) =m(p, t)g(Xe(p, t), t) onR (0, T),

(3.6)

8/14/2019 bb-sinum-1

7/28


and the initial condition

m(p, 0) = 0(p) in.(3.7)

We consider the standard weak formulation associated with this pure Lagrangianstrong problem:

m(p, t)m(p, t)(p)det F(p, t) dp +

Am(p, t)m(p, t) (p) dp

+ Rm(p, t)m(p, t)(p) dAp=

fm(p, t)(p)det F(p, t) dp

+

Rm(p, t)gm(p, t)(p) dAp,

(3.8)

H1D() and t(0, T). These are formal computations, i.e., we have assumedappropriate regularity on the involved data and solution.

4. Time discretization. In this section we introduce a second order scheme fortime semi-discretization of (3.8). We consider the general case where the diffusiontensor depends on the space variable and can degenerate, and the velocity field isnot divergence-free. Moreover, mixed Dirichlet-Robin boundary conditions are alsoallowed instead of merely Dirichlet ones.In the first part, we propose a time semi-discretization of (3.8) assuming that the

characteristic curves are exactly computed. Next, we propose a second-order Runge-Kutta scheme to approximate them and obtain some properties. Finally, stability anderror estimates are rigorously stated.

4.1. Second order semidiscretized scheme with exact characteristic curves.

We introduce the number of time steps, N, the time step t= T /N, and the mesh-pointstn= ntfor n = 0, 1/2, 1, . . . , N . Throughout this work, we use the notationn(y) :=(y, tn) for a function (y, t).The semi-discretization scheme we are going to study is a Crank-Nicholson-like scheme.It arises from approximating the material time derivative at t = tn+ 1

2, for 0 n

N1, by a centered formula and using a second order interpolation formula involvingvalues att = tn and t = tn+1 to approximate the rest of the terms at the same timet= tn+ 1

2. Thus, from (3.8), we have

1

2

n+1m (p)det F

n+1(p) +nm(p)det Fn(p) n+1m,t(p) nm,t(p)

t (p) dp

+1

4

An+1m (p) +Anm(p)n+1m,t(p) + nm,t(p) (p) dp+

4

R

mn+1(p) +mn(p) n+1m,t(p) +nm,t(p) (p) dAp=

1

2

det Fn+1(p)fn+1m (p) + det F

n(p)fnm(p)

(p) dp

+1

2

R

mn+1(p)gn+1m (p) +mn(p)gnm(p) (p) dAp.

(4.1)

Remark 4.1. In Section 4.4 we will prove that the approximations involved in

scheme (4.1) are O(t2) at point (p, tn+ 12

). Moreover, this order does not changeif we replace the exact characteristic curves and gradients F by accurate enoughapproximations.

8/14/2019 bb-sinum-1

8/28


4.2. Second order semidiscretized scheme with approximate charac-

teristic curves. In most cases, the analytical expression for motionXe is unknown;instead, we know the velocity field v. Let us assume thatXe(p, 0) = pp. Then,the motion Xe, assuming it exists, is the solution to the initial-value problem

Xe(p, t) = vm(p, t) Xe(p, 0) = p.(4.2)

Since the characteristics Xe(p, tn) cannot be exactly tracked in general, we proposethe following second order Runge-Kutta scheme to approximate Xne, n

{0, . . . , N

}.

Forn = 0:

X0RK(p) :=p p,(4.3)

and for 0 n N 1 we define by recurrence,

Xn+1RK(p) := XnRK(p) + tvn+

12 (Yn (p)) p,(4.4)

being

Yn (p) := XnRK(p) +t

2 vn(XnRK(p)).(4.5)

A similar notation to the one in

2 is used for the Jacobian tensor ofXnRK, namely,

F0RK(p) = I ,(4.6)

and for 0 n N 1,

Fn+1RK (p) = Fn

RK(p) + tLn+ 1

2 (Yn (p))

I+

t

2 Ln(XnRK(p))

FnRK(p).(4.7)

Now, we state some lemmas concerning properties of the approximate characteristicsXnRK. For this purpose, we require the time step to be upper bounded and thefollowing assumption:

Hypothesis 1. There exists a parameter > 0, such that the velocity field v isdefined in

T := t[0,T]

t {t},(4.8)

being

t :=

xt

B(x, ).(4.9)

Moreover v C1(T).Lemma 4.1. Under Hypothesis 1, there exists a parameter > 0 such that if

t < then XnRK(p) is definedp andn {0, . . . , N }, and the followinginclusion holds

XnRK() tn .

8/14/2019 bb-sinum-1

9/28


Proof. The result can be easily proved by applying Taylor expansion to Xe in thetime variable and using the regularity ofv.

Lemma 4.2. Under Hypothesis 1, ift < then

||FnRK||,eT(||L||,T+Ct) n {0, . . . , N }.(4.10)Here constantCdepends onv.

Proof. Inequality (4.10) follows by applying norms to (4.7), using the initial

condition (4.6) and applying the discrete Gronwall inequality.Lemma 4.3. Under Hypothesis 1 ift < min{, 1/(2||L||,T)}, then

||(FnRK)1||, eT(||L||,T+Ct) n {0, . . . , N }(4.11)and

(Fn+1RK )1(p) = (FnRK)

1(p)

I tLn+ 12 (Yn(p)) +O(t2)

,(4.12)

being the termO(t2) depending onv andp, and0 nN 1.Proof. Firstly, we can write

Fn+1RK (p) = MnRK(p)F

nRK(p),(4.13)

with

MnRK(p) :=I+ tLn+ 1

2 (Yn (p))

I+

t

2 Ln(XnRK(p))

.(4.14)

Now, by applying norms to (4.14) we have that||I MnRK||, < 1. Thus, MnRK(p)is invertible for 0 nN 1 and then, by induction, we deduce thatFn+1RK (p) is in-vertible too, with (Fn+1RK )

1(p) = (FnRK)1(p)(MnRK)

1(p). Moreover, (MnRK)1(p) =

j=0(I MnRK(p))j so (4.12) follows. The proof of (4.11) is analogous to the one ofthe previous lemma.The following corollaries can be easily proved (see [6] for further details).

Corollary 4.4. Under the assumptions of Lemma 4.2, we have

|| det Fn

RK||,eT(|| divv||

,T

+C(v)t)

,(4.15)det FnRK(p)> 0 ift < K,(4.16)

withKdepending onv and0 nN. Moreover,p det Fn+1RK (p) satisfies

det Fn+1RK (p) = det Fn

RK(p)

1 + t div vn+12 (Yn (p)) +O(t2)

,(4.17)

beingO(t2) depending onv and0 nN 1.Corollary 4.5. Under the assumptions of Lemma 4.3, we have

det(Fn+1RK )1(p) = det (FnRK)

1(p)

1 t div vn+ 12 (Yn (p)) +O(t2)

,(4.18)

p

,

n

{0, . . . , N

1

}, with O(t2) depending on v. Moreover,

n

{0, . . . , N }, we have

|| det(FnRK)1||,eT(||divv||,T+C(v)t).(4.19)

8/14/2019 bb-sinum-1

10/28


Lemma 4.6. Under Hypothesis 1 ift 0,

Cj > 0, j = 1, 2, constants depending on v andT, and u Rd with

|u|= 1.Proof. The result follows from expressions (4.10), (4.11), (4.15), (4.16) and (4.19),and by using the following equality

1 =|u|=(FnRK)T(p)(FnRK)T(p)u uRd,|u|= 1.(4.21)

Now, we consider a motion satisfying the following assumption:Hypothesis 2. The motionXe satisfies

t = Xe(p, t) = p pt [0, T].

Remark 4.2. Notice that, if the motion verifies Hypothesis 2 then

t = , v(x, t) = 0

x

t

[0, T].

Under Hypothesis 2, Lemma 4.1 can be improved.Lemma 4.7. Let us assume Hypothesis 2. Ift

8/14/2019 bb-sinum-1

11/28


Hypothesis 4. The diffusion tensor, A, is defined inO and belongs toW1,(O).Moreover,A is symmetric and has the following form:

A=

An1

,(4.26)

with An1 being a positive definite symmetric n1n1 tensor (n1 1) and anappropriate zero tensor. Besides, there exists a strictly positive constant, , which isa uniform lower bound for the eigenvalues ofAn1 .

Remark 4.3. Notice that the diffusion tensor can be degenerate in some applica-tions. This is the case, for instance, in some financial models where, nevertheless, thediffusion tensor satisfies Hypothesis 4.

Hypothesis 5. Functionf is defined inT and it is continuous with respect to thetime variable, in space L2.

Hypothesis 6. Function g is defined inTR and it is continuous with respect tothe time variable, in space H1. Besides, coefficient in boundary condition (2.9) isstrictly positive.Let us define the following sequences of functions ofp.AnRK := (FnRK)1A XnRK(FnRK)T det FnRK,

mnRK =|(FnRK)Tm| det FnRK,

for 0

n

N. Since usually the characteristic curves cannot be exactly computed,we replace in (4.1) the exact characteristic curves and gradient tensors by accurateenough approximations,

1

2

Xn+1RK det Fn+1RK + XnRKdet FnRK

n+1m,t nm,tt

dp

+1

4

An+1RK +AnRK n+1m,t+ nm,t dp+

4

R

mn+1RK +mnRK n+1m,t+nm,t dAp=

1

2

det Fn+1RK f

n+1 Xn+1RK + det FnRKfn XnRK

dp

+1

2 R mn+1RKgn+1 Xn+1RK +mnRKgn XnRK dAp.

(4.27)

For these computations we have made the assumptions of Lemma 4.3, and Hypothesis3, 4, 5 and 6.Notice that we have used a lowest order characteristics approximation formula pre-serving second order time accuracy.

Let us introduceLn+ 12t [] (H1()) andFn+ 1

2

t (H1()) defined byLn+ 12t [],

:=


2

n+1 nt

,

+

An+1RK +

AnRK

2

n+1 + n

2 ,

+

mn+1RK +mnRK2

n+1 +n

2

,

R

,

8/14/2019 bb-sinum-1

12/28


Fn+ 12t ,

:=

det Fn+1RK f

n+1 Xn+1RK + det FnRKfn XnRK2

,

+

mn+1RKgn+1 Xn+1RK +mnRKgn XnRK2

,

R

,

for C0(H1()) and H1().Remark 4.4. Regarding the definitions ofLn+ 12t [] andF

n+ 12

t , only the values offunction at discrete time steps

{tn

}Nn=0 are required. Thus, the above definitions

can also be stated for a sequence of functions= {n}Nn=0[H1()]N+1.Then the semidiscretized time scheme can be written as follows:

Given 0m,t, findm,t ={nm,t}Nn=1

H1D()

Nsuch that

Ln+ 12t [m,t],

=Fn+ 12t ,

H1D() for n = 0, . . . , N 1.

(4.28)

Remark 4.5. The stability and convergence properties to be studied in the nextsections still remain valid if we replace the approximation of characteristics appearingin scheme (4.28) by higher order ones or by the exact value.

4.3. Stability of the semidiscretized scheme. In order to prove stabilityestimates for problem (4.28), the assumptions considered in the previous section arerequired.

Firstly, we notice that, as a consequence of Hypothesis 4, there exists a unique positivedefinite symmetric n1 n1 tensor field, Cn1 , such that An1 = (Cn1)2. Let us denotebyC the symmetric and positive semidefinite d d tensor defined by

C=

Cn1

.(4.29)

Notice that A = C2 and C W1,(O). Let us denote by G the matrix withcoefficientsGij = | grad Cij|, 1 i, jd. At this point, let us introduce the constant

cA= max{||G||2, O , ||C||

2

, O },(4.30)

and the sequence of tensor fields

CnRK :=C XnRK(FnRK)Tdet FnRK n {0, . . . , N }.Next, let us denote byB thed d tensor

B=

In1

,(4.31)

whereIn1 is the n1 n1 identity matrix. Clearly, under Hypothesis 4 we have||Bw||2 Aw, w w Rd.(4.32)

Let us introduce the sequence of tensor fields

BnRK :=B(FnRK)Tdet FnRK n {0, . . . , N }.As far as the velocity field is defined inT (see Hypothesis 1), we can introduce thefollowing assumption:

8/14/2019 bb-sinum-1

13/28


Hypothesis 7. The velocity field satisfies

(I B)L(x, t)B = 0 (x, t) T.(4.33)

Remark 4.6. Hypothesis 7 is equivalent to having a velocity field v whosed n1last components depend only on the last d n1 variables.

Remark 4.7. For anyd dtensorEof the form given in (4.26) it is easy to checkthat

EHTw1, w2= EHTBw1, Bw2,for any d d tensor Hsatisfying (I B)HB = 0, and vectors w1, w2 Rd. Thisequality will be used below without explicitly stated. Moreover, under Hypothesis 7,if t < min{, 1/(2||L||,T)} it is easy to prove that

|B(FnRK)T(p)w| D|Bw|,forp , w Rd, n = 0, . . . , N , and D depending on v and T.Now, it is convenient to notice that Hypothesis 4 also covers the nondegenerate case.This hypothesis is usual in ultraparabolic equations (see, for instance, [24]), whichrepresent a wide class of degenerate diffusion equations arising from many applications(see, for instance, [5]). Furthermore, as stated in [19], ultraparabolic problems eitherhave C solutions or can be reduced to nondegenerate problems posed in a lowerspatial dimension. This is an important point, as the stability and error estimateswill be obtained under regularity assumptions on the solution.

In what follows, cv denotes the positive constant

cv := maxt[0,T]

||v(, t)||1,,t ,(4.34)

where|| ||1,,t is the norm given in (2.12). Moreover,Cv (respectively, J and D)will denote a generic positive constant, related to the norm of the velocity field v(respectively, to the rest of the data of the problem), not necessarily the same at eachoccurrence.

Corresponding to the semidiscretized scheme, we have to deal with sequences of

functions ={n}Nn=0. Thus, we will consider the spaces of sequences l(L2())andl2(L2()) equipped with their respective usual norms:

l(L2()):= max

0nN||n||,

l2(L2())

:=

t Nn=0

||n||2.(4.35)

Similar definitions are considered for functional spaces l(L2(R)) and l2(L2(R))associated with the Robin boundary condition and for vector-valued function spacesl(L2()) andl2(L2()). Moreover, let us introduce the notations

S[] :={n+1 +n}N1n=0, Rt[] :=

n+1 n

t

N1n=0

.

We denote by f XRKand by g XRK the following sequences of functionsf XRK :={fn XnRK}Nn=0 , g XRK :={gn XnRK}Nn=0 .

8/14/2019 bb-sinum-1

14/28


Before establishing some technical lemmas, let us recall the Youngs inequality

ab 12

a2 +

1

b2

,(4.36)

fora, b R and >0, which will be extensively used in what follows.Lemma 4.8. Let us assume Hypotheses 1, 3 and 4. Let{nm,t}Nn=1be the solution

of (4.28). Then, there exist a positive constant c(v, T , ) such that, fort < c, wehave Ln+ 12t [m,t], n+1m,t+nm,t

1t

Xn+1RK det Fn+1RK n+1m,t2

1t

XnRKdet FnRKnm,t2

+1

4

Cn+1RK n+1m,t+ nm,t2

+1

4

CnRKn+1m,t+ nm,t2

(4.37)

+

4

mn+1RK +mnRKn+1m,t+nm,t2R

c

det Fn+1RK

n+1m,t

2

+

det FnRKnm,t

2

,

wherec= 1,(cv+Cvt)/andn {0, . . . , N 1}.Proof. First, we decompose

Ln+ 12t [m,t], n+1m,t+nm,t

=I1+I2+I3, with

I1 =


2

n+1m,t nm,tt

, n+1m,t+nm,t

,

I2 =1

4

An+1RK +AnRK n+1m,t+ nm,t , n+1m,t+ nm,t

,

I3 =

4

mn+1RK +mnRK n+1m,t+nm,t , n+1m,t+nm,tR

.

LetKbe the constant appearing in Corollary 4.4. If t < K, we first have

I1 = Xn+1RK det Fn+1RK + XnRKdet FnRK2

n+1

m,t n

m,tt

, n+1m,t+nm,t

= 1

2t


12t

XnRKdet FnRKnm,t2

+ 1

2t

XnRKdet FnRKn+1m,t2

12t

Xn+1RK det Fn+1RK nm,t2

,

(4.38)

where we have used Hypothesis 3. Next, we introduce the function YnRK(p, ) :[tn, tn+1] tn , defined by YnRK(p, s) := XnRK(p)(tns)vn+

12 (Yn (p)), which

satisfies YnRK(p, tn) = XnRK(p) and Y

nRK(p, tn+1) = X

n+1RK(p). If t is small enough,

it is easy to prove that YnRK

(p,)

tn

. By hypothesis, is a differentiable function,then by Barrows rule and the chain rule, the following identity holds:

(XnRK(p)) = (Xn+1RK(p)) n(p) for a.e.p ,(4.39)

8/14/2019 bb-sinum-1

15/28


where

n(p) :=

tn+1tn

grad (YnRK(p, s)) vn+12 (Yn (p)) ds for a.e.p,(4.40)

verifies|n(p)| 1,cvt.Then, by using (4.17), (4.18) and (4.39) in (4.38), we get

I1 1t


1t

XnRKdet FnRKnm,t2

(4.41)

1, (cv+Cvt)det Fn+1RKn+1m,t2

+ det FnRKnm,t2 .

ForI2 we use the fact that A = C2 being Ca symmetric tensor field. We obtain,

I2 :=1

4

An+1RK +AnRKn+1m,t+ nm,t , n+1m,t+ nm,t

(4.42)

= 1

4

Cn+1RK n+1m,t+ nm,t2

+1

4

CnRKnm,t+ nm,t2

.

ForI3 we have

I3 =

4

mn+1RK +

mnRK

n+1m,t+

nm,t

2

R.(4.43)

Then, by summing up (4.41), (4.42) and (4.43) we get inequality (4.37).Lemma 4.9. Let us assume Hypotheses 1, 3, 4 and 7. Let{nm,t}Nn=1 be the

solution of (4.28) and >0 be the constant appearing in the Robin boundary condition(2.9). Then, there exist a positive constantc(v, T , ) such that, fort < c, we have

Ln+ 12t [m,t], n+1m,t nm,t

12t

Xn+1RK det Fn+1RK + XnRKdet FnRK n+1m,t nm,t2

+1

2

Cn+1RKn+1m,t2

12

CnRKnm,t2

(4.44)

+

2 mn+1RK

n+1m,t

2

R

2 mnRK

nm,t

2

R

ct Bn+1RKn+1m,t2

+ BnRKnm,t2

ctmn+1RKn+1m,t2

R+mnRKnm,t2

R

,

wherec= max {cACv /, Cv} andn {0, . . . , N 1}.Proof. First, we decompose

Ln+ 12t [m,t], n+1m,t nm,t

=I1+I2+I3, with

I1 =


2

n+1m,t nm,tt

, n+1m,t nm,t

,

I2 = 14 An+1RK +AnRK n+1m,t+ nm,t , n+1m,t nm,t ,I3 =

4

mn+1RK +mnRK n+1m,t+nm,t , n+1m,t nm,tR

.

8/14/2019 bb-sinum-1

16/28


ForI1, we use Hypothesis 3 to get

I1 = 1

2t


,

(4.45)

where we have assumed that t < K, being Kthe constant appearing in Corollary4.4. ForI2 we first have

I2 =1

4 Cn+1RKn+1m,t2 14 CnRKnm,t2(4.46)

+1

4

CnRKn+1m,t2

14

Cn+1RKnm,t2

.

Then we use Corollary 4.5, Hypotheses 4 and 7, and equality (4.7) to get

1

4

CnRKn+1m,t2

14

C XnRK(Fn+1RK )Tn+1m,tdet Fn+1RK 2

(4.47)

cACvt Bn+1RKn+1m,t2

.

Moreover, sinceAn1 is symmetric and positive definite,Cn1 =

An1 is a differentiabletensor field. Then by Barrows rule and the chain rule, the following identity holds,

C(Xn+1RK(p)) = C(XnRK(p)) +D

n(p) for a.e. p ,(4.48)where we have denoted by Dn thed d symmetric tensor field defined by

Dnij(p) :=

tn+1tn

grad Cij (YnRK(p, s)) vn+

12 (Yn (p)) ds,(4.49)

being YnRK the mapping defined in the proof of Lemma 4.8. Notice that D is of theform given in (4.29) and verifies||Dn||, cvcAt. Then, from the previousproperties, we have

1

4

CnRKn+1m,t

2

1

4

Cn+1RKn+1m,t

2

cACvt

Bn+1RKn+1m,t

2

.(4.50)

Similarly, we obtain the estimate

14|| Cn+1RKnm,t||2 14 || CnRKnm,t||2 cACvt|| BnRKnm,t||2.(4.51)

Thus, by introducing (4.50) and (4.51) in equality (4.46) we obtain the followinginequality:

I2 12

Cn+1RKn+1m,t2

12

CnRKnm,t2

(4.52)

cACvt Bn+1RKn+1m,t2

cACvt

BnRKnm,t2

.

ForI3 we first have

I3 =

4 mn+1RK

n+1m,t

2

R

4 mnRKnm,t

2

R

(4.53)

+

4

mnRKn+1m,t2R

4

mn+1RKnm,t2R

.

8/14/2019 bb-sinum-1

17/28


Next, by applying Corollaries 4.4, 4.5, Lemma 4.3 and equality (4.7) we obtain

I3 2

mn+1RKn+1m,t2R

2

mnRKnm,t2R

(4.54)

Cvtmn+1RKn+1m,t2

R+mnRKnm,t2

R

.

Then, by summing up (4.45), (4.52) and (4.54), inequality (4.44) follows.Now, in order to get error estimates we need to prove stability inequalities for moregeneral right-hand sides, namely for the problem,

Given 0m,t, findm,t = {nm,t}Nn=1

H1D()

Nsuch that

Ln+ 12t [m,t],

=Hn+ 12t ,

H1D() forn = 0, . . . , N 1,

(4.55)

withHn+ 12t ,

=

Sn+1,

+

Gn+1, R

.

Hypothesis 8.S= {Sn}Nn=1 [L2()]N andG= {Gn}Nn=1[L2(R)]N.Lemma 4.10. Let us assume Hypotheses 1 and 8. Let us suppose > 0 and

t < min{, 1/(2||L||,T), K}, being andKthe constants appearing, respectively,in Lemma 4.1 and in Corollary 4.4. Then, we have

Sn+1, ++ Gn+1, +Rcs||Sn+1||2+

1

2

det Fn+1RK 2

+det FnRK2

+

4cg

||Gn+1||2R

+

32

mn+1RK +mnRK(+)2R

,

(4.56)

, H1(), withcs = 1/c1 andcg = 1/(c1 c2), wherec1 andc2 are the constantsappearing in Lemma 4.6.

Proof. The estimate follows directly by applying the Cauchy-Schwarz inequalityto the left-hand side of (4.56), and using inequality (4.36) and Lemma 4.6.

Theorem 4.11. Let us assume Hypotheses 1, 3, 4 and 8. Letm,t be the solu-tion of (4.55) subject to the initial value0m,tH1D() and >0 be the constantappearing in the Robin boundary condition (2.9). Then there exist two positive con-stantsJ andD, which are independent of the diffusion tensor, such that ift < Dthen

det FRKm,tl(L2())

+

4

BRKS[m,t]l2(L2())

+

8

S[mRK]S[m,t]l2(L2(R))

J||0m,t||+||S||l2(L2())+ ||G||l2(L2(R)) .

(4.57)

whereS={Sn}Nn=1,G={Gn}Nn=1.

8/14/2019 bb-sinum-1

18/28


Proof. Sequence m,t ={nm,t}Nn=0 satisfiesLn+ 12t [m,t], n+1m,t+nm,t

=

Hn+ 12t , n+1m,t+nm,t

. We can use Lemma 4.8 to obtain a lower bound of this

expression, and Lemma 4.10 for = n+1m,t and = nm,t to obtain an upper bound.

By jointly considering both estimates, we get

1

t


1t

XnRKdet FnRKnm,t

2

+ 14 CnRKn+1m,t+ nm,t2

+

8mn+1RK +mnRKn+1m,t+nm,t2

R

cs||Sn+1||2+4cg

||Gn+1||2R

+cdet Fn+1RK n+1m,t2

+det FnRKnm,t2

,

(4.58)

wherec= max {1/, 21,(cv+Cvt)/}. Let us introduce the notation1n:=

det FnRKnm,t

2

,

2

n:=

4

n1

s=0 t BsRKs+1m,t+ sm,t2

,

n:=

8

n1s=0

t

ms+1RK+msRKs+1m,t+sm,t2R

.

Now, for a fixed integer q 1, let us sum (4.58) multiplied by t from n = 0 ton= q 1. Then, with the above notation we have

(1 ct)1q + 2q+ q 2ct q1n=0

1n+

10+ ||S||2l2(L2())+ ||G||2l2(L2(R)) ,where we have used Hypotheses 3 and 4. In the above equation denotes a positive

constant andc = max {1/, 21,(cv+ Cv t)/}. For t small enough, we canapply the discrete Gronwall inequality (see, for instance, [23]) and take the maximunin q {1, . . . , N }. Then, estimate (4.57) follows.

Corollary 4.12. Let us assume Hypotheses 1, 3, 4, 5 and 6. Letm,t be thesolution of (4.28) subject to the initial value0m,tH1D(). Then, there exist twopositive constants J and D, independent of the diffusion tensor and such that, fort < D, we have

det FRKm,tl(L2())

+

4

BRKS[m,t]l2(L2())

+

8

S[mRK]S[m,t]

l2(L2(R))

J

||0m,t||

+ det FRKf XRKl2(L2())

+ mRKg XRKl2(L2(R))

.(4.59)

8/14/2019 bb-sinum-1

19/28


Proof. The result follows directly by replacing

Sn+1 with 1/2

det Fn+1RK fn+1 Xn+1RK + det FnRKfn XnRK

andGn+1 with 1/2

mn+1RKgn+1 Xn+1RK +mnRKgn XnRK in (4.57).Lemma 4.13. Let us assume Hypotheses 1 and 8. Let t < min{, K}, being

and K the constants appearing in Lemma 4.1 and in Corollary 4.4, respectively.Then, we have

Sn+1, 2cst ||Sn+1||2+ 16t det Fn+1RK + det FnRK( )2

,

(4.60)

, L2(), wherecs is the constant appearing in Lemma 4.10.

Proof. The result easily follows by applying the Cauchy-Schwarz inequality, in-equality (4.36) with = 8t/and Lemma 4.6.

Lemma 4.14. Let us assume Hypotheses 1 and 8. Suppose that > 0 andt < min{, 1/(2||L||,T), K}. Then, for any sequence{n}Nn=0 [L2(R)]N+1and anyq {1, . . . , N }, the following inequality holds:

q1

n=0Gn+1, n+1 nR 4cg

||Gq

||2

R+

16 ||mqRKq||2R + 1

2 ||G1

||2

R

+

2||0||2R+

tcg2

q1n=1

Gn+1 Gnt2R

+t

2

q1n=1

||mnRKn||2R .

(4.61)

Proof. The result follows from the equality

q1n=0

Gn+1, n+1 nR =Gq, qR G1, 0R(4.62)

t

q1

n=1Gn+1

Gn

t , nR .

Indeed, the three terms on the right-hand side can be bounded by using the Cauchy-Schwarz inequality, inequality (4.36) and Lemma 4.6.

Theorem 4.15. Let us assume Hypotheses 1, 3, 4, 7 and 8, and letm,t bethe solution of (4.55) subject to the initial value 0m,t H1D(). Let > 0 bethe constant appearing in the Robin boundary condition (2.9). Then, there exist twopositive constantsJ(v, cA/, T)andD(, v, T , cA/) such that ift < D then

4

S[det FRK]Rt[m,t]l2(L2())

+

2

BRKm,tl(L2())

+

4

mRKm,tl(L2(R)) J

2 B0m,t+

4

0m,tR+ ||S||l2(L2())+ ||G||l(L2(R))+ Rt[G]l2(L2(R))

.

(4.63)

8/14/2019 bb-sinum-1

20/28


Proof. Sequence m,t ={nm,t}Nn=0 satisfiesLn+ 12t [m,t], n+1m,t nm,t

=

Hn+ 12t , n+1m,t nm,t

. Then, we use Lemma 4.9 and Lemma 4.13 for = n+1m,tand= nm,tto obtain, respectively, a lower and an upper bound for this expression.By jointly considering both estimates, we get

1

2t


+1

2

Cn+1RKn+1m,t2

12

CnRKnm,t2

+

2

mn+1RKn+1m,t2R

2

mnRKnm,t2R

ct Bn+1RKn+1m,t2

+ BnRKnm,t2

+ctmn+1RKn+1m,t2

R+mnRKnm,t2

R

+

2cst

||Sn+1||2

+

16t

det Fn+1RK + det Fn

RK(n+1m,t nm,t)

2

+

Gn+1, n+1m,t nm,tR

,

(4.64)withc= max {cACv/, Cv}. Forn = 0, . . . , N , let us introduce the notations

1n :=

4t

n1s=0

det Fs+1RK + det FsRKs+1m,t sm,t2

,

2n:=

2

BnRKnm,t2

, n:=

4

mnRKnm,t2R

.

Now, for a fixed q 1, let us sum (4.64) from n = 0 to n = q 1. With the abovenotation and by using Lemma 4.14 for=

m,t, we get

1q+ (1 2ct)2q + (1 4ct)q 4ct q1n=0

2n+ 10ct q1n=0

n

+

20+ 0+ ||S||2l2(L2())+ ||G||2l(L2(R))+ Rt[G]2

l2(L2(R))

,(4.65)

where we have used Hypotheses 3 and 4. In the above equationc= max {cACv/, Cv}and denotes a positive constant. For t small enough, we can apply the discreteGronwall inequality (see, for instance, [23]) and take the maximun in q {1, . . . , N }.Thus, estimate (4.63) follows.

Corollary 4.16. Let us assume Hypotheses 1, 3, 4, 5, 6 and 7, and letm,tbe the solution of (4.28) subject to the initial value0m,t H1D(). Then there exist

8/14/2019 bb-sinum-1

21/28


two positive constantsJ(v, cA/, T)andD(, v, T , cA/) such that ift < D then

4


+

2

BRKm,tl(L2())

+

4

mRKm,tl(L2(R))

J

2

B0m,t+4 0m,tR+ det FRKf XRKl2(L2()) + mRKg XRKl(L2(R))

+ Rt[mRKg XRK]

l2(L2(R))

.

(4.66)

Proof. The result follows directly by replacing

Sn+1 with 1/2

det Fn+1RK fn+1 Xn+1RK + det FnRKfn XnRK

andGn+1 with 1/2

mn+1RKg Xn+1RK +mnRKg XnRK in (4.63).Remark 4.8. Notice that, constants J and D appearing in Theorem 4.15 and

Corollary 4.16 depend on the diffusion tensor, more precisely they depend on fractioncA

. In most cases this fraction is bounded in the hyperbolic limit.

Remark 4.9. In the particular case of Dirichlet boundary conditions (D ),diffusion tensor of the formA= B andf= 0, the previous corollary can be improved.Specifically, by using analogous procedures to the ones in the previous corollary wecan obtain the followingl(H1) stability result with constants (Jand D) independentof the diffusion constant

2


+

1

2

BRKm,tl(L2())

J(1 + )

1

2

B0m,t .(4.67)

for t < D.

4.4. Error estimate for the semidiscretized scheme. The aim of the presentsection is to estimate the difference between thediscrete solutionof (4.28), m,t:=

{nm,t}Nn=0, and the exact solution of the continuous problem,m:={nm}Nn=0. Ac-cording to (3.8) fortn+ 1

2, with 0n N 1, the latter solves the problem

Ln+ 12 [m], = Fn+ 12 , H1D(),(4.68)whereLn+ 12 [m](H1()) andFn+ 12 (H1()) are defined by

Ln+ 12 [

m],

:=

Xn+ 12e det Fn+ 12

mn+ 1

2

,

+ An+ 12m n+ 12m ,

+mn+ 12 n+ 12m , R

,Fn+ 12 ,

:=

det Fn+12 fn+

12 Xn+ 12e ,

+mn+ 12 gn+ 12 Xn+ 12e ,

R,

8/14/2019 bb-sinum-1

22/28


H1().The error estimate in the l(L2())-norm, to be stated in Theorem 4.27, is provedby means of Theorem 4.11 and the forthcoming Lemmas 4.25 and 4.26. On the otherhand, the error estimate for the gradient in the l(L2())-norm, to be stated inTheorem 4.28, is proved by means of Theorem 4.15 and the forthcoming Lemmas4.25 and 4.26. Before doing this we give some results with sketched proofs (see [6] forfurther details). Some auxiliary mappings will be introduced. They will be denotedby , u and depending on whether they are scalar, vector or tensor mappings,

respectively. Moreover, if v is smooth enough, it is easy to prove that F, F1

,det Fand their partial derivatives, as well as the ones of (FnRK)1 and det FnRK can

be bounded by constants depending only on v and T. These estimates and the onesobtained in 4.2 forFnRK, (FnRK)1 and det FnRKwill be used below without explicitlystated (see [6] for further details).

Lemma 4.17. Let us assume Hypotheses 1 and 3. Let us suppose that vC2(T), XeC4( [0, T]), t < , C3(L2()) andmC2(L()). Let usdefine the functionn+

12 : R, forn {0, . . . , N 1}, by

n+12 (p) := Xn+ 12e (p)det Fn+ 12 (p) n+ 12 (p)1

2

Xn+1RK(p)det Fn+1RK (p) + XnRK(p)det FnRK(p)

n+1(p) n(p)t

,

for a.e. p . Then n+ 1

2 L2

() and||n+ 1

2 || C(T, v, )t2

||||C3(L2()),n= 0, . . . , N 1.

Proof. The result follows by using Taylor expansions and noting that if XeC3( [0, T]) and v C1(T) then|Xne(p) XnRK(p)| C(v, T)t2, and ifXeC4( [0, T]) and v C2(T) then| det Fn(p) det FnRK(p)| C(v, T)t2.

Lemma 4.18. Let us assume that Am C2(L()). Let w C2(L2()) be agiven mapping andun+

12 : Rd, forn {0, . . . , N 1}, be defined by

un+12 (p) :=An+ 12m (p)wn+ 12 (p) An+1m (p) +Anm(p)

2

wn+1(p) + wn(p)

2

,

for a.e. p

. Then, un+

12

L2() and

||un+

12

||

C(T, v, A)t2

||w

||C2(L2()),

n {0, . . . , N 1}. Moreover, ifXeC4( [0, T]), AmC2(W1,())and wC2(H1()) then un+

12 H1() and|| Div un+ 12 || C(T, v, A)t2||w||C2(H1()),

n {0, . . . , N 1}.

Proof. The result follows by writing Taylor expansions in the time variable for wand the tensor fieldAm(p, s) := det F(p, s)F1(p, s)A Xe(p, s)FT(p, s), s [0, T].

Lemma 4.19. Let us assume Hypotheses 1 and 4. Let us suppose that vC2(T), Xe C4( [0, T]) andt < min{, 1/(2||L||,T)}, being the constantappearing in Lemma 4.1. Let w L2() be a given function and un : Rd bedefined by

un(p) :=Anm(p)w(p) AnRK(p)w(p), 0 nN.(4.69)Then, un L2() and||un|| C(T, v, A)t2||w||. Moreover, if v C3(T),Xe C5( [0, T]), A W2,(O) and w H1(), then un H1() and

8/14/2019 bb-sinum-1

23/28


|| Div un||C(T, v, A)t2||w||1,2,.

Proof. The result follows by applying Taylor expansions, noting that if XeC4( [0, T]) and v C2(T) then||Xne XnRK||1,, C(v, T)t2,||(Fn)T (FnRK)

T||, C(v, T)t2 and|| det Fn det FnRK||, C(v, T)t2, and ifXeC5( [0, T]) and v C3(T) then||(Fn)T (FnRK)T||1,,C(v, T)t2and|| det Fn det FnRK||1,,C(v, T)t2.

Lemma 4.20. Let

C2(L2(R)) be a given mapping and

n+ 12

1 : R

R,

n+12

2 : R R be defined by

n+ 1

2

1 (p) :=mn+ 12 (p)n+ 12 (p) mn+1(p) +mn(p)2

n+1(p) +n+1(p)

2

,

n+ 1

2

2 (p) :=mn+ 12 (p)n+ 12 (p) mn+1(p)n+1(p) +mn(p)n(p)2

.

Thenn+ 1

2

1 , n+ 1

2

2 L2(R) and

||n+ 121 ||Rt2

8 ||

m||C2(L2(R))C(T, v)t2||||C2(L2(R)),

||

n+ 12

2

||R

C(T, v)t2

||||

C2(L2(R)).

Proof. The result follows by using Taylor expansions in the time variable.Lemma 4.21. Let us assume Hypothesis 1. Let us suppose that v C2(T),

XeC4( [0, T])andt < min{, 1/(2||L||,T)}, being the constant appearingin Lemma 4.1. Let L2(R) be a given function andn : R R be defined by

n(p) :=mn(p)(p) mnRK(p)(p), 0n N.(4.70)Thenn L2(R) and||n||R C(T, v)t2||||R.

Proof. The result follows noting that| det Fn(p) det FnRK(p)| C(v, T)t2and

(Fn)T(p) (FnRK)T(p)

2C(v, T)t2.

Lemma 4.22. Let us assume Hypotheses 1. Let us suppose v

C2(T

), XeC4( [0, T]) and t < min{, 1/(2||L||,T)}, being the constant appearing in

Lemma 4.1. Let H1(Gtn)be a given function, beingGtn the set defined in (4.24),and letn : R R be defined by

n(p) :=mn(p)(Xne(p)) mnRK(p)(XnRK(p)), 0 nN.(4.71)Thenn L2(R) and||n||R C(T, v)t2||||1,2,Gtn .

Proof. The result follows by applying Taylor expansions, noting that|Xne(p) XnRK(p)| C(v, T)t2,|(Fn)T(p) (FnRK)T(p)| C(v, T)t2 and| det Fn(p) det FnRK(p)| C(v, T)t2.

Lemma 4.23. Let C2(L2()) be a given function and n+ 12 : R, forn {0, . . . , N 1}, be defined by

n+12 (p) := det Fn+

12 (p)n+

12 (p) det F

n+1(p)n+1(p) + det Fn(p)n(p)

2 .

8/14/2019 bb-sinum-1

24/28


Then,n+12 L2() andn+ 12

t

2

8 || det F ||C2(L2())C(T, v)t2||||C2(L2()).

Proof. The result follows by applying Taylor expansions.Lemma 4.24. Let us assume Hypothesis 1. Let us suppose that v C2(T),

Xe C4([0, T]) and t < , being the constant appearing in Lemma 4.1.Let H

1

(

tn) be a given function, being

tn the set defined in (4.9), and letn : R be defined byn(p) := det Fn(p)(Xne(p)) det FnRK(p)(XnRK(p)), 0 nN.

Thenn L2()and||n|| C(T, v)t2||||1,2,tn .

Proof. The result follows by using Taylor expansions, noting that|Xne(p)XnRK(p)| C(v, T)t2 and|det Fn(p) det FnRK(p)| C(v, T)t2.

Lemma 4.25. Assume Hypotheses 1, 3 and 4 hold. Moreover, suppose thatXeC5( [0, T])and that the coefficients of problem (2.7)-(2.10) satisfy,

v C3(T), m C2(L()), AW2,(O), Am C2(W1,()).

Let the solution of (4.68) satisfy,mC3(L2()), m C2(H1()), m|RC2(L2(R)).

Finally, assume that t < min{, 1/(2||L||,T)}. Then, for each0 n N 1,there exist two functions

n+ 12

L: R andn+ 12L : R R, such that

Ln+ 12 Ln+ 12t

[m], = n+ 12L , + n+ 12L , R ,(4.72)H1D(). Moreover,

n+ 12

L L2(), n+ 12L L2(R) and the following estimates

hold:

n+ 1

2

L t2C(T, v, , A) ||m||C3(L2())+ ||m||C2(H1()) ,(4.73) n+ 12L R t2C(T, v, A) ||m m||C2(L2(R))+||m||C2(L2(R)) ,

where >0 appears in (2.9).

Proof. The left-hand side of (4.72) is equal to I1+I2+I3, with

I1 =

Xn+ 12e det Fn+ 12

mn+ 1

2

,

1

2


n+1m nmt

,

,

I2 = An+ 1

2m

n+ 12

m

An+1RK +

AnRK

2 n+1m + nm

2 , I3 =

mn+ 12 n+ 12m mn+1RK +mnRK2

n+1m +

nm

2

,

R

.

8/14/2019 bb-sinum-1

25/28


The bound for I1 directly follows from Lema 4.17 for = m, so we can define a

functionn+ 1

2

I1 L2() such that

I1 =

n+ 1

2

I1,

, withn+ 12I1 C(T, v, )t2||m||C3(L2()).(4.74)

In order to estimate I2 we apply Lemmas 4.18 and 4.19 for w =m and w =n+1m + nm, respectively, so I2 =

u

n+ 12

I2,

, with u

n+ 12

I2 H1(). Then, by

using a Greens formula, we deduce

I2 =

un+ 1

2

I2 m, R

Div un+ 1

2

I2,

,

where, the involved functions are bounded as follows:un+ 12I2 mR C(T, v, A)t2||m m||C2(L2(R)),(4.75) Div un+ 12I2 C(T, v, A)t2||m||C2(H1()).The estimate for I3 follows by applying Lemmas 4.20 and 4.21 for = m|R and= (n+1m +

nm)|R , respectively:

I3 = n+ 12I3 , R with n+ 12I3 R C(T, v)t2||m||C2(L2(R)).(4.76)Finally, partial results (4.74), (4.75) and (4.76) imply (4.72).

Lemma 4.26. Assume Hypothesis 1, and v C2(T), Xe C4([0, T])andt 0.

Hypothesis 10. Functions appearing in problem (2.7)-(2.10) satisfy:

m C2(L()),A W2,(O), AmC3(W1,()), v C3(T), fm C2(L2()),f C1(T), gm C3(L2(R)),gC2(TR) and >0.

8/14/2019 bb-sinum-1

26/28


Lemmas in this section hold under Hypotheses 1, 3 and 4 and the previous ones.

Theorem 4.27. Assume Hypotheses 1, 3, 4, 5, 6, 7 and 9, and Xe C5( [0, T]). Let

mC3(L2()), m C2(H1()), m|RC2(L2(R)),

be the solution of (4.68) and letm,t be the solution of (4.28) subject to the initial

value0

m,t = 0

m =0

H1

(). Then, there exist two positive constantsJ andD,the latter being independent of the diffusion tensor, such that, ift < D we have

||

det FRK(m m,t)||l(L2())+

4

BRKS[m m,t]l2(L2())

+

8

S[ mRK]S[m m,t]l2(L2(R))

J t2 ||m||C3(L2())+||m||C2(H1())+ ||m m||C2(L2(R))+ ||m||C2(L2(R))

+|| det F fm||C2(L2())+ ||f||C1(T)+ ||

mgm||C2(L2(R))+ ||g||C1(T

R)

.

(4.79)

Proof. We denote by em,tthe difference between the continuous and the discrete

solution, that is, em,t =

nm nm,tN

n=0. Then, by using (4.28) and (4.68) we have

Ln+ 12t [em,t],

=

Ln+ 12t Ln+12

[m], + Fn+ 12 Fn+ 12t , ,(4.80)

H1D() and 0 n N 1. Then, as a consequence of Lemmas 4.25 and 4.26,we deduce

Ln+ 1

2

t [em,t], = n+ 1

2

f

n+ 12

L, +

n+ 12

g

n+ 12

L, R ,(4.81)

H1D(). Now the result follows by applying Theorem 4.11 to (4.81), notingthate0m,t = 0 and using the upper bounds for L, f, L andg given in Lemmas4.25 and 4.26.

Remark 4.10. Notice that constant J appearing in the previous theorem isbounded in the limit when the diffusion tensor vanishes. In particular, Theorem4.27 is also valid when A 0.

Theorem 4.28. Let us assume Hypotheses 1, 3, 4, 5, 6, 7 and 10, and XeC5( [0, T]). Letm with

mC3(L2()), m C3(H1()), m|RC3(L2(R)),

be the solution of (4.68) andm,t be the solution of (4.28) subject to the initial value0m,t =

0m =

0 H1(). Then, there exist two positive constantsJ andD such

8/14/2019 bb-sinum-1

27/28


that, fort < D we have

4

S[det FRK]Rt[m m,t]l2(L2())

+

2

BRK(mm,t)l(L2())

+

4

mRK(m m,t)l(L2(R)) J t2 ||m||C3(L2())+||m||C2(H1())+ ||m m||C3(L2(R))+ ||m||C3(L2(R))

+|| det F fm||C2(L2())+ ||f||C1(T)+ ||mgm||C3(L2(R))+ ||g||C2(TR

)

.

(4.82)

Proof. It is analogous to the one of the previous theorem, but using Theorem 4.15instead of Theorem 4.11 and noting that Rt[L ]

l2(L2(R))+ Rt[g]

l2(L2(R))Ct2 ||m m||C3(L2(R))

+ ||m||C3(L2(R))+ ||

mgm||C3(L2(R))+ ||g||C2(T

R)

.

This estimate follows by using Taylor expansions andXn+1e (p) Xn+1RK(p) (Xne(p) XnRK(p)) Ct3,(Fn+1)1(p) (Fn+1RK )1(p) (Fn)1(p) (FnRK)1(p) Ct3, det Fn+1(p) det Fn+1RK (p) (det Fn(p) det FnRK(p)) Ct3.Remark 4.11. In the particular case of diffusion tensor of the form A = B with

>0, constants JandD appearing in the previous theorem are bounded as 0.Remark 4.12. Notice that, from the obtained estimates and by using a change

of variable, we can deduce similar ones in Eulerian coordinates (see [6] for furtherdetails).

5. Conclusions. We have performed the numerical analysis of a second-orderpure Lagrangian method for convection-diffusion equations with degenerate diffusiontensor and non-divergence-free velocity fields. Moreover, we have considered generalDirichlet-Robin boundary conditions. The method has been introduced and analyzedby using the formalism of continuum mechanics. Although our analysis considers anyvelocity field and use approximate characteristic curves, second order error estimateshave been obtained when smooth enough data and solutions are available. In thesecond part of this paper ([7]), we analyze a fully discretized pure Lagrange-Galerkinscheme and present numerical examples showing the predicted behavior (see also [6]).

REFERENCES

[1] R.A. Adams. Sobolev Spaces. Academic Press, New York, 1975.[2] J. Baranger, D. Esslaoui, and A. Machmoum. Error estimate for convection problem with

characteristics method. Numer. Algorithms, 21 (1999):4956. Numerical methods for par-tial differential equations (Marrakech, 1998).

8/14/2019 bb-sinum-1

28/28


[3] J. Baranger and A. Machmoum. Une norme naturelle p our la methode des caracteristiquesen elements finis discontinus: cas 1-D. RAIRO Model. Math. Anal. Numer., 30:549574,1996.

[4] J. Baranger and A. Machmoum. A natural norm for the method of characteristics usingdiscontinuous finite elements: 2D and 3D case. Math. Model. Numer. Anal., 33:12231240,1999.

[5] E. Barucci, S. Pilodoro, and V. Vespri. Some results on partial differential equations and Asianoptions. Math. Models Methods Appl. Sci., 11:475497, 2001.

[6] M. Bentez. Metodos numericos para problemas de conveccion difusion. Aplicacion a la con-veccion natural, Ph.D. Thesis, Universidade de Santiago de Compostela. 2009.

[7] M. Bentez and A. Bermudez. Numerical Analysis of a second-order pure Lagrange-Galerkinmethod for convection-diffusion problems. Part II: fully discretized scheme and numericalresults. Submitted.

[8] A. Bermudez and J. Durany. La methode des characteristiques pour les problemes deconvection-diffusion stationnaires. Math. Model. Numer. Anal., 21:726, 1987.

[9] A. Bermudez, M.R. Nogueiras, and C. Vazquez. Numerical solution of (degenerated)convection-diffusion-reaction problems with higher order characteristics/finite elements.Part I: Time discretization. SIAM. J. Numer. Anal., 44:18291853, 2006.

[10] A. Bermudez, M.R. Nogueiras, and C. Vazquez. Numerical solution of (degenerated)convection-diffusion-reaction problems with higher order characteristics/finite elements.Part II: Fully Discretized Scheme and Quadrature Formulas. SIAM. J. Numer. Anal.,44:18541876, 2006.

[11] K. Boukir, Y. Maday, and B. Metivet. A high-order characteristics method for the incompress-ible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg., 116:211218, 1994.ICOSAHOM92 (Montpellier, 1992).

[12] K. Boukir, Y. Maday, B. Metivet, and E. Razafindrakoto. A high-order characteristics/finiteelement method for the incompressible Navier-Stokes equations. Internat. J. Numer. Meth-ods Fluids, 25:14211454, 1997.

[13] K. Chrysafinos and N. J. Walkington. Error estimates for discontinuous Galerkin approxima-tions of implicit parabolic equations. SIAM J. Numer. Anal., 43:24782499 (electronic),2006.

[14] K. Chrysafinos and N. J. Walkington. Error estimates for the discontinuous Galerkin methodsfor parabolic equations. SIAM J. Numer. Anal., 44:349366 (electronic), 2006.

[15] K. Chrysafinos and N. J. Walkington. Lagrangian and moving mesh methods for the convectiondiffusion equation. M2AN Math. Model. Numer. Anal., 42:2555, 2008.

[16] J. Douglas, Jr., and T.F. Russell. Numerical methods for convection-dominated diffusion prob-lems based on combining the method of characteristics with finite element or finite differ-ence procedures. SIAM J. Numer. Anal., 19:871885, 1982.

[17] R.E. Ewing and H. Wang. A summary of numerical methods for time-dependent advection-dominated partial differential equations. J. Comput. Appl. Math., 128:423445, 2001.

[18] M.E. Gurtin. An Introduction to Continuum Mechanics. Mathematics in Science and Engi-neering, 158, Academic Press, San Diego, 1981.

[19] L. Hormander. Hypoelliptic second order differential equations. Acta Math., 119:147171, 1967.

[20] K. W. Morton. Numerical solution of convection-diffusion problems.Appl. Math. Comput., 12,Chapman Hall, London, 1996.

[21] O. Pironneau. On the Transport-Diffusion Algorithm and Its Applications to the Navier-StokesEquations. Numer. Math., 38:309332, 1982.

[22] O. Pironneau, J. Liou, and T. Tezduyar. Characteristic-Galerkin and Galerkin/least-squaresspace-time formulations for the advection-diffusion equations with time-dependent do-mains. Comput. Methods Appl. Mech. Engrg., 100:117141, 1992.

[23] A. Quarteroni and A. Valli. Numerical approximation of partial differential equations.SpringerSeries in Computational Mathematics, Springer-Verlag, 23, Berlin, 1994.

[24] M. A. Ragusa. On weak solutions of ultraparabolic equations. Nonlinear Anal., 47:503511,2001.

[25] H. Rui and M. Tabata. A second order characteristic finite element scheme for convection-diffusion problems. Numer. Math., 92:161177, 2002.

[26] P. Wilmot, J. Dewynne, and S. Howison. Option pricing. mathematical models and computa-tion. Oxford Financial Press, Oxford, 1993.

bb-sinum-1

Documents