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www.elsevier.com/locate/apnum

Error estimates for finite volume element methods for convection–diffusion–reaction equations

Rajen K. Sinha

a,

, Jürgen Geiser

b

aDepartment of Mathematics, Indian Institute of Technology Guwahati, Guwahati 781039, India bWeierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D-10117 Berlin, Germany

Available online 19 January 2006

Abstract

In this paper, we study finite volume element (FVE) method for convection–diffusion–reaction equations in a two-dimensional convex polygonal domain. These types of equations arise in the modeling of a waste scenario of a radioactive contaminant transport and reaction in flowing groundwater. Both spatially discrete scheme and discrete-in-time scheme are analyzed in this paper. For the spatially discrete scheme, optimal order error estimates inL2andH1norms are obtained for the homogeneous equation using energy method. Further, a quasi-optimal order error estimate inLnorm is shown to hold in an interior subdomain away from the corners. Based on backward Euler method, a time discretization scheme is discussed and related error estimates are derived.

©2005 IMACS. Published by Elsevier B.V. All rights reserved.

MSC:65M60; 65N30; 65N15

Keywords:Convection–diffusion–reaction equation; Finite volume element method; Spatially discrete scheme; Discrete-in-time scheme; Error estimates

1. Introduction

Our mathematical formulation is based on a potential waste scenario of radioactive contaminants, which are trans- ported and reacted with flowing groundwater in porous media (cf. [11–13]). The model is described in the formulation as an initial-boundary value problem of the form

ut+ ∇ ·

vuD(x)u

+λu=f (x, t ) inΩ×J (1.1)

subject to the boundary conditions

u=g1(x, t ) onΓ1; (vuD∇u)·n=g2(x, t ) onΓ2 (1.2) and initial condition

u(x,0)=u0(x) inΩ. (1.3)

* Corresponding author.

E-mail addresses:rajen@iitg.ernet.in (R.K. Sinha), geiser@wias-berlin.de (J. Geiser).

0168-9274/$30.00©2005 IMACS. Published by Elsevier B.V. All rights reserved.

doi:10.1016/j.apnum.2005.12.002

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Here,Ω ⊂R2 is a bounded convex polygonal domain with boundaryΓ =Γ1Γ2,J =(0, T] withT <∞ and ut=∂u/∂t. Further,D= {di,j(x)}is a symmetric and uniformly positive definite dispersion–diffusion matrix of size 2×2 inΩ. The parametervis the divergence free groundwater velocity andλis the constant reaction parameter. The nonhomogeneous termf and the coefficientsdij(x)are assumed to be smooth for our purpose.

With the substitutionu˜=ug1onΓ1, we rewrite Eqs. (1.1)–(1.3) as

˜

ut+ ∇ ·

vu˜−D(x)∇ ˜u

+λu˜=f (x, t ) inΩ×J (1.4)

subject to the boundary conditions

˜

u=0 onΓ1; (vu˜−D∇ ˜u)·n=g2 onΓ2 (1.5)

and initial condition

˜

u(x,0)=u0(x) inΩ. (1.6)

Thus, study of problem (1.1)–(1.3) now reduces to the study of equivalent problem (1.4)–(1.6).

In the recent years, the use of finite volume element methods has become popular due to its certain conservation feature that are desirable in many applications (cf. [8–10]). The FVE method considered in these paper are based on Petrov–Galerkin formulation in which solution space consisting of continuous piecewise polynomial functions and the test space consisting of piecewise constant functions. The test space essentially conserve the local conservation property of the method. In [8,9], the authors have studied this type of problem with self-adjoint elliptic operator and proved optimalL2andH1error estimates which requires higher regularity requirement on the solution when compared to that of finite element method (cf. [17,20]). Recently, the authors of [6] have studied FVE for self-adjoint parabolic problem with homogeneous Dirichlet boundary condition and derived optimal error estimate inL2andH1 norms, and suboptimal order of error estimate inLnorm. They have used semigroup theory in a crucial way in their analysis.

In this present paper, we study the convergence of FVE methods for a non-selfadjoint parabolic problem. Both spa- tially discrete scheme and discrete-in-time scheme are discussed and optimal error estimates inL2andH1norms are proved using only energy method. In addition, a quasi-optimal order inLnorm is obtained in an interior subdomain away from the corners. Our analysis avoid the use of semigroup theory and the regularity requirement on the solution is same as that of finite element method. Further, based on backward Euler method the fully discrete scheme is ana- lyzed and related optimal error estimates are established. To the best our knowledge error estimates for the problem (1.1)–(1.3) using FVE method have not been established earlier.

The literature on the theoretical framework and the basic tools for the analysis of the finite volume element methods for elliptic and parabolic problems are described in [3–5,7,10,15,16,18,19] and references therein.

A brief outline of this paper is as follows. In Section 2, we introduce some notations and present some preliminary materials to be used in our subsequent sections. The Petrov–Ritz projection is introduced and related estimates are carried out in Section 3. Section 4 is devoted to the error estimates for the FVE method. Finally, the backward Euler time discretization scheme is discussed in Section 5.

Throughout this paper,Cdenotes a generic positive constant which does not depend on the spatial and time dis- cretization parametershandk, respectively.

2. Notations and preliminaries

LetV = {φH1(Ω)|φ=0 onΓ1}. For the purpose of finite volume element approximation of (1.4)–(1.6), the weak formulation of the problem may be stated as follows: Findu˜:J¯→V such that

(u˜t, φ)+A(u, φ)˜ = g2, φ +(f, φ),φV (2.1)

withu(0)˜ =u0, where the bilinear formA(·,·)is given by

A(u, φ)˜ =

Ω

D(x)∇ ˜u· ∇φvu˜∇φ+λuφ˜

dx. (2.2)

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Here and below, we denote(·,·)and · byL2inner product and the induced norm onL2(Ω). The notation ·,·is used to denote boundary integral over Γ2. Further, we shall use the standard notation for Sobolev spacesWm,p(Ω) with 1p∞. The norm onWm,p(Ω)is defined by

um,p,Ω= um,p=

Ω

|α|m

Dαupdx 1/p

, 1p <

with the standard modification forp= ∞. Whenp=2, we writeWm,2(Ω)byHm(Ω)and denote the norm by·m. For a fractional numbers, Sobolev spaceHs is defined in [1].

Note that the bilinear form A(·,·)given by (2.2) may not be coercive but it can be made coercive by adding a sufficiently large constantκ∈Rtimes theL2-inner product. That is, it satisfies Gärding’s type inequality (cf. [2])

A(φ, φ)+κφ2α

2φ21,φV .

Introducing the transformationu¯=eκtu˜as a new dependent variable, we rewrite (1.4) as

¯

ut+Aκu¯= ¯f =eκtf, tJ (2.3)

withu(0)¯ =u0,where Aku¯= ∇ ·

vu¯−D(x)∇ ¯u

++κ)u.¯

The weak form corresponding to (2.3) is defined to be the functionu¯:J¯→V such that

(u¯t, φ)+Aκ(u, φ)¯ = ¯g2, φ +(f , φ),¯ ∀φV (2.4) withg¯2=e−κtg2andu(0)¯ =u0. The bilinear formAκ(·,·)is given by

Aκ(u, φ)¯ =

Ω

D(x)∇ ¯u· ∇φdx−

Ω

vu¯∇φdx+

Ω

+κ)uφ¯ dx. (2.5)

2.1. A priori estimates

Following the lines of proof in [17], it is easy to derive a priori bounds for the solutionu¯ satisfying (2.3) under appropriate regularity assumption on the initial functionu0. The details are thus omitted.

Lemma 2.1.Letu0L2(Ω)andg2H1/22). Then, forf =0, we have u(t )¯ 2+

t

0

u(s)¯ 2

1dsC

u02+ t

0

g22H1/22)ds

.

Moreover, whenu0V, we have u(t ) ¯ 2

1+ t

0

u¯s(s) 2+ u(s) ¯ 2

2

dsC

u021+ t

0

g22H1/22)ds

.

Lemma 2.2.Assume thatu0H2(Ω)V,∂tjgj2H1/22) (j=0,1,2)andf=0. Then, we have u¯t(t ) 2+

t

0

u¯s(s) 2

1dsC

u022+ 1 j=0

t

0

jg2

∂tj 2

H1/22)

ds

,

t u¯t(t ) 2

1+ t

0

s ¯uss2dsC

u022+ 1 j=0

t

0

jg2

∂tj 2

H1/22)

ds

,

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t2 u¯t t(t ) 2+ t

0

s2 u¯ss(s) 2

1dsC

u022+ 2 j=0

t

0

jg2

∂tj 2

H1/22)

ds

,

ti iu¯

∂ti

2

C

u02+ i

j=0

t

0

jg2

∂tj 2

H1/22)

ds 1/2

, i=0,1, t∈J.

2.2. Finite volume element approximation

LetThbe a quasi-uniform triangulation ofΩsuch thatΩ=

KThK, whereKis a closed triangle element. Let Nhbe the set of all nodes or vertices ofTh, i.e.,

Nh= {p: pis a vertex of elementKThandpΩ}.

Further, we denoteNh0=NhΩ. For a vertexxiNh, letΠ (i)be the index set of those vertices that, along withxi, are in some element ofTh.

For the triangulation Th, we now introduce a dual meshTh as follows: In each elementKTh consisting of verticesxi,xj andxk, select a pointqK, and select a pointxij by straight linesγij,K. Then, for a vertexxi, we let Vi be the polygon whose edges areγij,K in whichxi is a vertex of the elementK. We call thisVi acontrol volume centered atxi. Further, we note that

xiNhVi= Ω. Thus, the dual meshThis then defined as the collection of these control volumes. Acontrol volumecentered at a vertexxiis given in Fig. 1.

We call the control volume meshThregular or quasi-uniform if there exists a positive constantC >0 such that C1h2meas(Vi)Ch2 for allViTh,

wherehis the maximum diameter of all elementsKTh.

There are various ways to introduce a regular dual meshThdepending on the choices of the pointq in an element KTh and the pointsxij on its edges. In this paper, we chooseq to be the barycenter of an elementKTh, and the pointsxij are chosen to be the midpoints of the edges ofK. In addition, ifTh is locally regular, i.e., there is a constantCsuch that

Ch2Kmeas(K)h2K,

wherehK=diam(K)for all elementsKTh. Then the dual meshThis also locally regular. For the purpose of finite volume element approximation letShbe the linear finite element space defined on the triangulationTh,

Sh=

vC(Ω): v|K is linear for allKThandv|Γ1=0 ,

and its dual volume element spaceSh, Sh=

vL2(Ω): v|V is constant for allVThandv|Γ1=0 .

Fig. 1. Control volumes with barycenter as internal point and interfaceγijofViandVj.

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Obviously,Sh=span{φi(x): xiNh0}andSh=span{χi(x): xiNh0}, whereφi are the standard nodal basis func- tions associated with the nodexi, andχi are the characteristic functions of the volumeVi. LetIh:C(Ω)Shand Ih:C(Ω)Shbe the usual interpolation operators, i.e.,

Ihu¯=

xiNh

¯

uiφi(x) and Ihu¯=

xiNh

¯ uiχi(x),

whereu¯i= ¯u(xi).

The FVE approximation corresponding to (2.4) is defined to be the functionu¯h(t ):J¯→Shsuch that

(u¯h,t, Ihχ )+Aκ(u¯h, Ihχ )= ¯g2, Ihχ +(f , I¯ hχ ) (2.6) for allχShwithu¯h(0)=u0,h, whereu0,his a suitable projection ofu0ontoShto be defined later.

The bilinear formsAκ(·,·)in (2.6) is defined by

Aκ(u, w)¯ =

xiNh

wi

∂Vi

D(x)∇ ¯uvu¯

·ndSx+wi

Vi

+κ)u¯dx

for(u, w)¯ ∈((VH2)Sh)×Sh, wherenis the outer-normal vector of the involved integration domain. Note that when(u, w)¯ ∈V ×V the bilinear formAκ(·,·)is given by (2.5). Similarly, the FVE approximation to (2.1) is easily obtained by takingκ=0 in (2.6).

In order to describe features of the bilinear forms defined in (2.4) and (2.6) we define some discrete norms onSh andSh,

|uh|20,h=(uh, uh)0,h, |uh|21,h=

xiNh

xj∈Π (i)

meas(Vi)

(uhiuhj)/dij2

,

uh21,h= |uh|20,h+ |uh|21,h, |||uh||| =(uh, Ihuh), where(uh, vh)0,h=

xiNhmeas(Vi)uhivhi=(Ihuh, Ihvh)anddij =d(xi, xj)is the distance between xi andxj. These norms are well defined foruhShas well anduh0,h= |||uh|||.

Below, we state the equivalence of the discrete norms|·|0,hand·1,hwith usual norms·and·1, respectively onSh. Further, some properties of the bilinear forms are stated without proof. For a proof, we refer to [9,10].

Lemma 2.3.There exist two positive constantsC1andC2such that for allvhSh, we have C1|vh|0,hvhC2|vh|0,h,

C1|||vh|||vhC2|||vh|||, C1vh1,hvh1C2vh1,h.

Lemma 2.4.There exist positive constantsCandcsuch that, for allφh, ψhSh, the boundedness property Aκh, Ihψh)Cφh1ψh1

and the coercive property

Aκh, Ihφh)cφh21

hold true.

The following lemma gives the key feature of the bilinear forms in the finite volume element method. For a proof, see [10].

Lemma 2.5.Letφ(VH2)Sh. Then we have

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Aκ(φ, χ )Aκ(φ, Ihχ )=

KTh

K

−∇ ·(D∇φvφ)++κ)φ

Ihχ )dx

+

KTh

∂K

(D∇φvφ)·n

Ihχ )dS, ∀χSh.

3. Petrov–Ritz projection and related estimates

Following [8,9], define the Petrov–Ritz projectionRh:VH2(Ω)Shby

Aκ(u¯−Rhu, I¯ hχ )=0, ∀χSh. (3.1)

The following lemma prove to be convenient for obtainingH1andL2error estimates for the Petrov–Ritz projection.

Lemma 3.1.Assume thatφShandDW2,(Ω). Then we have Aκ(φ, χ )Aκ(φ, Ihχ )Chφ1χ1,χSh. Further, forφVH2(Ω), we have

Aκ(φ, χ )Aκ(φ, Ihχ )Chφ2χ1,χSh.

Proof. Since the dual mesh is formed by the barycenters, we have forχSh

K

Ihχ )dx=0 for allKTh.

Thus, in view of Lemma 2.5, we have forφ, χSh

Aκ(φ, χ )Aκ(φ, Ihχ )=

KTh

K

−∇ ·(D∇φvφ)++κ)φ

Ihχ )dx

+

KTh

∂K

(DDK)(φvφ)·n

Ihχ )dS

:=I1+I2. (3.2)

Here,DK is a function designed in a piecewise manner such that for any edgeE of a triangleKTh andxE, DK(x)=D(xc), wherexcis the mid point ofE. Noting that, forφSh,∇φis a constant onK, we have∇ ·(D∇φ)= (∇ ·D)φ. Now, applying Cauchy–Schwarz’s inequality and using the fact thatχIhχChχ1, we obtain

|I1|Chφ1χ1. (3.3)

Since|D(x)DK|hD1,andχIhχL2(∂K)Ch1/2χ1,K(cf. [10]), the termI2is bounded by

|I2|Ch

KTh

h1/2∇φL2(∂K)χ1,KCh

KTh

φ1,Kχ1,KChφ1χ1, (3.4)

where in the second inequality we have used the fact that∇φis constant onK. Combine (3.2) and (3.4) to prove the first inequality.

Next, forφVH2(Ω), we have

|I1|Chφ2χ1. (3.5)

ForI2, using the trace theorem [2], we obtain

|I2|Ch

K∈Th

h1/2φL2(∂K)χIhχL2(∂K)Chφ2χ1. (3.6)

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Combine (3.2), (3.5) and (3.6) to obtain the second inequality and this completes the proof. 2 Setρ= ¯uRhu. We now establish¯ H1-error estimate forρand its temporal derivative.

Lemma 3.2.Letρsatisfy(3.1). Then we have ρ1Ch ¯u2, ρt1Ch ¯ut2.

Proof. Withφh=Ihu¯−Rhu, we obtain using (3.1)¯ 21Aκ(ρ , ρ)

=Aκ(ρ ,u¯−Ihu)¯ +Aκ(ρ , Ihu¯−Rhu)¯

=Aκ(ρ ,u¯−Ihu)¯ +Aκ(ρ , φh)Aκ(ρ , Ihφh).

An application of Lemma 3.1 yields Aκ(ρ , φh)Aκ(ρ , Ihφh)=

Aκ(u, φ¯ h)Aκ(u, I¯ hφh)

Aκ(Rhu, φ¯ h)Aκ(Rhu, I¯ hφh)

Ch

¯u2+ ¯u1

φh1

Ch ¯u2

ρ1+h ¯u2

,

where in the last inequality we have usedφh1C(h ¯u2+ ρ1). Thus, we obtain 21Ch ¯u2ρ1+Ch2 ¯u22.

Kickback the termρ1to obtain the first inequality. For the second inequality, differentiate (3.1) with respect to time t to have

Aκt, Ihχ )=0. (3.7)

Then the rest of the proof follows in a similar fashion. 2

We shall prove theL2estimates ofρand its temporal derivatives in the following theorem.

Lemma 3.3.Letρsatisfy(3.1). Then we have ρ(t ) Ch2 ¯u2, ρt(t ) Ch2 ¯ut2.

Proof. The proof will proceed by duality argument. LetψH2(Ω)H01(Ω)be the solution of

Aκψ=ρ inΩ, ψ=0 on∂Ω, (3.8)

whereAκis the formal adjoint ofAκ. The solutionψsatisfies the following regularity estimate

ψ2Cρ. (3.9)

Multiplying (3.8) byρand then takingL2inner-product overΩ, we obtain

ρ2=Aκ(ρ , ψIhψ )+Aκ(ρ , Ihψ )=I1+I2. (3.10) Using Lemma 3.2,I1is bounded as

|I1|Ch2 ¯u2ψ2. (3.11)

Following the line of arguments of [10, Theorem 3.5], the termI2is bounded as

|I2|Ch2u2ψ2 (3.12)

which combine with (3.10), (3.11) and (3.9) completes the proof. 2

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4. Error estimates for the spatially discrete scheme

In this section, the error analysis for the spatially discrete FVE approximation will be carried out. For homogeneous problem, optimal order error estimates are established inL2andH1norms whenu0H2V. In addition, a quasi- optimal order error estimate inLnorm is proved in an interior sub-domain away from the corners.

As usual we split the errore= ¯u− ¯uhas e=(u¯−Rhu)¯ +(Rhu¯− ¯uh)=ρ+θ .

Since the estimates ofρare already known, it is enough to have estimates forθ.

Using (2.6), an equation of the form (2.6) withuh replaced byuand (3.1), it is easy to verify thatθ satisfies an error equation

t, Ihχ )+Aκ(θ , Ihχ )= −t, Ihχ ),χSh. (4.1) Defineθ (t )ˆ =t

0θ (s)ds. Then, clearlyθ (0)ˆ =0 andθˆt=θ. We shall prove a sequence of lemmas which lead to the desired result.

Lemma 4.1.Assume thatu¯h(0)=Rhu0. There is a positive constantCindependent ofhsuch that t

0

θ (s) 2ds+ θ (t )ˆ 2

1C

t ρ(0) 2+ t

0

ρ(s) 2ds

.

Proof. Integrate (4.1) from 0 totand use the factθ (0)=0 to have (θ , Ihχ )+Aκ(θ , Iˆ hχ )= −

ρ(t ), Ihθ +

ρ(0), Ihθ

. (4.2)

Chooseχ=θin (4.2) to obtain

|||θ|||2+1 2

d dt

Aκ(θ ,ˆ θ )ˆ

= −(ρ , Ihθ )+

ρ(0), Ihθ +

Aκ(θ , θ )ˆ −Aκ(θ , Iˆ hθ )

ρ + ρ(0) θ +C ˆθ1θ, (4.3)

where in the last step, we have used the fact that (cf. [6, Lemma 4.1]) Aκ(θ , θ )ˆ −Aκ(θ , Iˆ hθ )C ˆθ1θ.

Integrating (4.3) from 0 totand using Lemma 2.3, we obtain t

0

θ (s) 2ds+ θ (t )ˆ 2

1C t

0

ρ2+ ρ(0) 2 ds+1

2 t

0

θ2ds+ t

0

ˆθ21ds.

Kickback the term 12t

0θ2dsand then apply Gronwall’s lemma to complete the rest of the proof. 2

Lemma 4.2.Letθsatisfy(4.1)withu¯h(0)=Rhu0. Then there is a positive constantC independent ofhsuch that

t θ (t ) 2+ t

0

s θ (s) 2

1dsC

t ρ(0) 2+ t

0

ρ(s) 2+s2 ρs(s) 2 ds

.

Proof. Setχ=t θ in (4.1). Then using the symmetry of(ψ, Ihχ ),ψ, χShonSh, we obtain 1

2 d dt

t|||θ|||2

+t Aκ(θ , Ihθ )|||θ|||2+tIhθ.

Integrating from 0 totand using the weak coercivity in Lemma 2.4, it now leads to

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1

2tθ (t )2+ t

0

s θ (s) 2

1dsC t

0

θ2ds+ t

0

tθds.

Apply Young’s inequality to have

t θ (t ) 2+ t

0

s θ (s) 2

1dsC t

0

θ2+s2ρs2 ds

.

Finally, use Lemma 4.1 to complete the rest of the proof. 2

Lemma 4.3.Let the hypotheses in Lemma4.2hold true. Then there is a positive constantC independent ofhsuch that

t

0

s2 θ (s) 2ds+t2 θ (t ) 2

1C

t ρ(0) 2+ t

0

ρ(s) 2+s2 ρs(s) 2 ds

.

Proof. Chooseχ=t2θt in (4.1) to have t2θt(t )2+1

2 d dt

t2Aκ(θ , θ )

= −t2t, Ihθt)+t Aκ(θ , θ )+t2

Aκ(θ , θt)Aκ(θ , Ihθt)

. (4.4)

It follows from [6, Lemma 4.1] that

Aκ(θ , θt)Aκ(θ , Ihθt)Cθ1θt.

Now integrate (4.4) from 0 tot. Then apply Lemmas 2.3 and 2.4 and standard kickback argument to obtain t

0

s2 θs(s) 2ds+t2 θ (t ) 2

1C t

0

s θ (s) 2

1ds+C t

0

s2ρs2ds+C t

0

s2θ21ds.

Finally, apply Lemma 4.2 and Gronwall’s lemma to complete the proof. 2 The main results of this section is given in the following theorems.

Theorem 4.1.Letu˜ satisfy(1.4)withf =0, and letu˜h be its FVE approximation. Then, foru0H2V, ∂tjgj2H1/22) (j=0,1,2)andu¯h(0)=Rhu0, we have

u(t )˜ − ˜uh(t )

1Cht1/2

u02+ 2

j=0

t

0

jg2

∂tj 2

H1/22)

ds 1/2

and

u(t )˜ − ˜uh(t ) Ch2

u02+ 2

j=0

t

0

jg2

∂tj 2

H1/22)

ds 1/2

hold true fortJ.

Proof. By triangle inequality, we have u(t )¯ − ¯uh(t )

1 ρ(t )

1+ θ (t )

1. From Lemma 4.3, we obtain

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1C

tRhu0u02+ t

0

ρ(s) 2+s2 ρs(s) 2 ds

1/2

Ch

tu022+ t

0

¯u22+s2 ¯us22 ds

1/2

.

In view of Lemma 2.2, it now follows that

1Cht1/2

u02+ 2

j=0

t

0

jg2

∂tj 2

H1/22)

ds 1/2

, (4.5)

and this together with Lemmas 3.2, 2.2 and the identity

¯

u− ¯uh=eκt(u˜− ˜uh) (4.6)

yield the first inequality. Similarly, for the second inequality, we use Lemmas 3.3, 4.2, a priori estimates in Lemma 2.2 and the identity (4.6). This completes the rest of the proof. 2

We shall close this section by showing a quasi-optimal order error estimate in maximum norm in an interior domain Ω0ΩwithΩ0not containing any vertex ofΩ.

Theorem 4.2.LetΩ0Ω be such thatΩ0does not contain any vertex ofΩ. Further, letu˜satisfy(1.4)withf =0, and letu˜h be its FVE approximation. Assume thatu0H2V, jg2

∂tjH1/22) (j =0,1,2)andu¯h(0)=Rhu0. Then there is a positive constantCsuch that

u(t )˜ − ˜uh(t )

L0)Ct1h2log1 h

u02+ 2

j=0

t

0

jg2

∂tj 2

H1/22)

ds 1/2

, tJ.

Proof. By triangle inequality, we have u(t )¯ − ¯uh(t )

L0) θ (t )

L0)+ ρ(t )

L0). (4.7)

Recall thatShis the linear finite element space and triangulation is quasi-uniform, we thus have (cf. [20, Chapter 5]) θ (t )

LC

log1 h

1/2

θ (t )1, and hence, using (4.5), it now follows that

θ (t )

L0)Ch2t1/2log1 h

u02+ 2

j=0

t

0

jg2

∂tj 2

H1/22)

ds 1/2

. (4.8)

Thus, the first term in (4.7) is bounded as desired. It now remains to boundρL0). LetΩ2andΩ3be domains withΩ1Ω2Ω3Ω and smooth boundaries. Further, letΩ3does not contain any corner ofΩand the distances between∂Ω3Ω,∂Ω2Ω, and∂Ω1Ωare positive. Letωbe a smooth function such thatω|Ω2=1 andω|∂Ω3Ω = 0. It is well known that (cf. [6])

ρ(t )

L0)Ch2log1 h u(t )¯

W2,∞2)+C ρ(t ) . (4.9)

Since the termρis bounded as desired by Lemma 3.3, it now remains to bound the first term ¯u(t )W2,∞2). Using Sobolev inequality and elliptic regularity estimate inΩ3(recall that∂Ω3is smooth), we obtain, withu¯¯=ωu,¯

¯uW2,∞2)C ¯uW3,p2)C ¯¯uW3,p3)CAκu¯¯W1,p3)

C

Aκu¯W1,p+ ¯uW2,p

CAκu¯W1,p, (4.10)

(11)

where 2< p <2/(2−β)with 1< β. In the last inequality, we have used the following regularity estimate (cf. [14, Theorem 5.2.7])

¯uW2,pCAκu¯Lp.

Using (2.3) withf=0, Sobolev inequality and Lemma 2.2, it now follows that Aκu¯W1,p C ¯utW1,p C ¯utH2

C ¯ut tCt1

u02+ 2

j=0

t

0

jg2

∂tj 2

H1/22)

ds 1/2

(4.11)

fortJ. Combine (4.7)–(4.11) with (4.6) to complete the rest of the proof. 2 5. Discrete-in-time scheme

In this section, based on backward Euler method we shall discuss fully discrete approximations to (2.6). While optimal order error estimates are obtained inL2andH1norms, a quasi-optimal order error estimate inLnorm is established in any sub-domain away from the corners.

Let k >0 be the time step and tn=nk withT =N k. For any continuous functionψ (t ), setψn=ψ (tn)and

¯tψn=k1nψn−1). ForφSh, defineφj,has

φj,h=sup

gSh

(φ, Ihg) gj

, j=0,1.

The discrete in time Euler scheme is to seek a functionUn,n=1,2, . . . , Nsatisfying ¯tUn, Ihχ

+Aκ

Un, Ihχ

= ¯g2, Ihχ +f¯n, Ihχ

χSh, (5.1)

with givenU0=Rhu0.

Set Un=eκtnUn, where Un is the backward Euler approximation to (1.4) which may be obtained by putting κ=0 in (5.1). Note that ifUn’s are known then we can easily computeUn’s.

Denoteηn=Un− ¯unh. Then, from (2.6) and (5.1),ηnsatisfies ¯tηn, Ihχ

+Aκ

ηn, Ihχ

=

τn, Ihχ

, χSh (5.2)

withη0=0, whereτn= ¯unht− ¯tu¯nh.

Lemma 5.1.Letηnsatisfy(5.2)andu¯h(0)=Rhu0. Then there exists a constantCindependent ofksuch that ηn 2+k

n

j=1

ηj 2

1Ck2

u022+ 1 j=0

t

0

jg2

∂tj 2

H1/22)

ds

.

Proof. Takingχ=ηnin (5.2) and using the symmetry of(χ , Ihψ ), χ , ψShonSh, and the identity(∂¯tηn, Ihηn)=

1

2¯t{|||ηn|||2} +k2|||¯tηn|||2leads to 1

2¯tηn2 +Aκ

ηn, Ihηn +k

2¯tηn2=

τn, Ihηn τn

1,h ηn

1. Apply Young’s inequality and kickback the termηn21to obtain

1

2¯tηn2

+ ηn 2

1C τn 2

1,h.

Summing overnfrom 1 tomand using Lemma 2.3, it now leads to ηm 2+k

m

n=1

ηn 2

1C

η0 2+k m

n=1

τn 2

1,h

.

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