additive schwarz methods for parabolic problems

Applied Mathematics and Computation 163 (2005) 17–28

www.elsevier.com/locate/amc

Additive Schwarz methodsfor parabolic problems

Jianhua Yang *, Danping Yang

School of Mathematics and System Sciences, Shandong University, Shanda Road, Shandong,

Jinan 250100, PR China

Abstract

Based on domain decomposition, we give an additive Schwarz domain decomposi-

tion method for semi-linear parabolic problems. We consider the dependence of con-

vergence rate of this algorithm on parameters of time step and space-mesh. We give the

error estimate, which tell us that the convergence of the approximate solution is inde-

pendent of the iteration number at each time level.

� 2004 Elsevier Inc. All rights reserved.

Keywords: Domain decomposition; Additive Schwarz method; Parabolic problems; Error estimate;

Convergence rate

1. Introduction

Additive Schwarz method, based on domain decomposition, is a powerful

iteration method for solving elliptic equations and other stationary problems. A

systematic theory has been developed for elliptic finite element problems in the

past few years, see [2,5]. In [3,4] Cai constructed some kinds of additive Schwarz

algorithms and multiplicative Schwarz methods and prove that the convergencerate is smaller than one for parabolic equations, however, there the author did

not consider the dependence of the convergence and the parameters. In [6,7] the

authors presented a non-overlapping domain decomposition method for

* Corresponding author.

E-mail address: [email protected] (J. Yang).

0096-3003/$ - see front matter � 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.amc.2004.03.025

mail to: [email protected]

18 J. Yang, D. Yang / Appl. Math. Comput. 163 (2005) 17–28

parabolic equations, but since they use explicit schemes at intersection points,

the stability condition Dt6 12H 2 is needed.

The outline of this paper is as follows. In Section 2 we formulate the additive

algorithms with backward Euler scheme in time variable and Galerkin

approximate in space variable. We also give the convergence rate. In Section 3

we establish some Lemmas about the properties of domain decomposition. In

Section 4 we give the proof of the theorem in Section 2.

Throughout this paper, L2ðXÞ and HrðXÞ denote the general Sobolev spaceswith norms k � k and k � kr respectively, c and C, with or without subscripts,denote generic positive constants. Their values may be different on differentoccasions, but are independent of the mesh parameter h and time stepping Dt.

2. Additive Schwarz methods and convergence results

Consider the following parabolic problem: find uðx; tÞ such that

ouot

�Xdi¼0

Xdj¼0

oðaijðxÞ ouoxiÞ

oxj¼ f ðx; tÞ in X � J ; ð1Þ

u ¼ 0 on oX � J ; ð2Þ

uðx; 0Þ ¼ u0ðxÞ in X; ð3Þ

where X is a bounded polygonal domain in R2 with boundary oX, J ¼ ð0; T �is the time interval. For all i; j aij ¼ aji and exists a positive constant c suchthat

Xdi¼1

Xdj¼1aijninj P cknk2 8n ¼ ðn1; n2ÞT 2 R2: ð4Þ

The standard variational formulation of the above problem is to find

uðtÞ 2 H 10 ðXÞ in t 2 J such that
ouot
; v� �

þ aðu; vÞ ¼ ðf ðuÞ; vÞ; v 2 H 10 ðXÞ; ð5Þ

ðuð0Þ; vÞ ¼ ðu0; vÞ; v 2 H 10 ðXÞ; ð6Þ

where

aðu; vÞ ¼Z

X

Xdi¼1

Xdj¼1aijninj

ouoxi

ovoxjdn; ð7Þ

ðf ðuÞ; vÞ ¼Z

Xf ðuÞvdn: ð8Þ

J. Yang, D. Yang / Appl. Math. Comput. 163 (2005) 17–28 19

We assume that the bilinear form is

(1) bounded, that is

jaðu; vÞj6 kukH10ðXÞkvkH1

0ðXÞ 8u; v 2 H 10 ðXÞ; ð9Þ

(2) elliptic, that is

aðu; vÞP ckukH10ðXÞ 8u 2 H 10 ðXÞ: ð10Þ

Let Dt be the time step, tn ¼ nDt, un ¼ uðtnÞ, we have

un � un�1

Dt¼ ou

otþO o2u

ot2Dt

� �; ð11Þ

then Eq. (1) can be approximated by,

un � un�1

Dt; v

� �þ aðu; vÞ ¼ ðf ðun�1Þ; vÞ þ ðqn; vÞ 8v 2 H 10 ðXÞ; ð12Þ

where

qn ¼ un � un�1

Dt� ou

otþ f ðunÞ � f ðun�1Þ ¼ O o2u

ot2Dt

� �:

For a given polygonal region X 2 R2, let Xi be a shape regular finite elementtriangulation of X and H the maximal diameter of these Xi’s. We sometimes

refer Xi as a substructure and fXig as the coarse mesh or H -level triangulationof X.We further divide each substructure Xi into smaller triangles, denoted as

sji ; j ¼ 1; . . . We assume that sji form a shape regular finite element triangula-tion of X and h is the maximal diameter of sji . We callfsjig the fine mesh or h-level triangulation of X.We next define the piecewise linear finite element function spaces over both

H -level and h-level triangulations of X.

V H ¼ fvH jcontinuous on X; vH jXi linear on Xi; 8i; vH ¼ 0 on oXg;

V h ¼ fvhjcontinuous on X; vhjsji linear on sji ; 8i; vh ¼ 0 on oXg:

It is obvious that V H � V h. We also use the notations

Kh ¼ fxj 2 interior nodes of h-level subdivisiong;

and

KH ¼ fxj 2 interior nodes of H-level subdivisiong:


To obtain a fully discrete problem, we discrete Eq. (12) in space by using a

Galerkin finite element method and the subspace V h � H 10 ðXÞ. The Galerkinapproximation of Eq. (12) reads as follows: Find unh 2 V h, such that

unh � un�1h

Dt; v

� �þ aðunh; vÞ ¼ ðf ðun�1h Þ; vÞ; 8v 2 V h; ð13Þ

ðu0h; vÞ ¼ ðu0; vÞ: ð14Þ

It is well known that there hold the error estimates

kun � unhks ¼ Oðhrþ1 þ DtÞ: ð15Þ

We first introduce our basic decomposition of X and the correspondingprojections. We first extend each subregion Xi to obtain X0

i, such that Xi � X0i

and there exists a constant a > 0 satisfying: distanceðoX0i \ X; oXi \ XÞP aH ,

8i. Suppose that oX0i dose not cut through any h-level elements. We make the

same constructions for the subregions that meet the boundary except that wecut off the parts that are outside X. To simplify the notations, we also denoteX00 ¼ X.It is easy to see that the finite element space V h can be decomposed into the

sum of the coarse mesh function space V H and a number of spaces which are

supported only in subregions Xi, i.e. V h ¼ V h0 þ V h

1 þ � � � þ V hN , where V

h0 ¼ V H

and V hi ¼ V h \ H 10 ðX

0iÞ.

Let Phi be the projection from V h to V hi with respect to the bilinear form

Að�; �Þ and Ph ¼ Ph0 þ Ph1 þ � � � þ PhN . We obtain the derived equation with re-spect to the bilinear form Að�; �Þ and the decomposition {V h

i }

Phuh ¼ g0n�1:

Here g0n�1 can be computed without a priori knowledge of the solution uh.

It can be seen easily that the operator P is symmetric with respect to A-norm.The standard conjugate gradient method in A-norm can therefore be used. Inthis paper we focus on the study of additive Schwarz iterative algorithms for

solving (13).

Scheme 1. For nP 1 find W n 2 Mh by

1. Set Wn0 ¼ W n�1.

2. For j ¼ 1; 2; . . . ;m when i ¼ 1; 2 . . . ;M find WjMþi satisfying

W njMþi � W n�1

Dt; v

!þ aðW n

jMþi; vÞ ¼ ðf ðW n�1Þ; vÞ 8v 2 V hi ; ð16Þ

W njMþi ¼ W

nðj�1ÞM ;

when i ¼ 0, find ðWjM � IHW ðj�1ÞMÞ 2 V h0 satisfying


W njM � W n�1

Dt; v

!þ aðW n

jM ; vÞ ¼ ðf ðW n�1Þ; vÞ 8v 2 V h0 ;

there IH is the interpolation operator,

W jM ¼ 1

M þ 1XMi¼0W njMþi:

3. Let W n ¼ W mM , x 2 X.Here m denotes the iteration number at each timelevel.

Let

Aðu; vÞ ¼ ðu; vÞ þ Dtaðu; vÞ 8v 2 H 10 ðXÞ;

kukA ¼ ðAðu; uÞÞ1=2 ¼ ðkuk20 þ Dtkuk2aÞ1=2

;

kuka ¼ ðaðu; uÞÞ1=2:

Theorem 1. Suppose that the solution of (1) is sufficiently smooth and that W bethe solution of Scheme 1 we have

kun � W nkA6C hkþ1�

þ Dt þ lnðH=hÞ1þ lnðH=hÞ

� �m�; ð17Þ

where C denotes a generic constant independent of Dt, h and H .

3. Some lemmas

In order to prove Theorem 1, in this section we give some auxiliary lemmas.

Lemma 1. There exist two constants c > 0 and C, which depend only on theregularity of the finite element subdivision of X, such that

chdXxi2Kh

ðuhðxiÞÞ26 kuhk2L2ðXÞ 6ChdXxi2Kh

ðuhðxiÞÞ2 8uh 2 V h:

The statement is also true if we replace V h by V H and h by H .

Lemma 2. Let Xi be a substructure, uh 2 V hi and let uhi ¼ 1

areaðXiÞR

Xiuh dX, then

kuh � uhi k6Cð1þ lnðH=hÞÞjuhj2H1ðXiÞ:


Lemma 3 (Lions). For any uh 2 V h, if there is a decomposition uh ¼PN

i¼0uhi ,

uhi 2 V hi and there exists a constant C0 such that

XNi¼0

kuhi k2

a6C0kuhk2

a 8uh 2 V h; ð18Þ

then we have, kmin (which is the lowest eigenvalue of Ph) satisfy

kminP1

C0: ð19Þ

Proof. From projection theorem

kuhk2a ¼ aðuh; uhÞ ¼XNi¼0aðuh; uhi Þ ¼

XNi¼0aðuh; Phi uhi Þ

¼XNi¼0aðPhi uh; uhi Þ6

XNi¼0

kPhi uhk2a

!1=2 XNi¼0

kuhk22

!1=2;

from the inequality that is given, for 8uh 2 V h

kuhk2a6C0XNi¼0

kPhi uhk2a ¼ C0

XNi¼0aðPhi uh; uhÞ ¼ C0aðPhuh; uhÞ;

so, we can get, kminP 1=C0. h

Lemma 4. For any uh 2 V h, there exists uhi 2 V hi ði ¼ 0; 1; . . . ;NÞ such that

uh ¼XNi¼0uhi ð20Þ

and there exists a constant C, which is independent of h and H , such that

XNi¼0

kuhi k2

A6Ckuk2

A: ð21Þ

Proof. For 8uh 2 V h there exists a decomposition

uh ¼ IHuh þ ðuh � IHuhÞ ¼ IHuh þ wh;

from unit decomposition theorem [1], we know that there exist some functions

hi 2 C10 ðX0

iÞ such that
XNi¼0
hi ¼ 1 06 hi6 1;


let uhi ¼ IhðhiuhÞ ði ¼ 1; 2; . . . ;NÞ, there Ih is interpolate operator, it is easily tobe seen uhi 2 V h

i and wh ¼PN

i¼1uhi , and let IHuh ¼ uh0 2 V h

0 , such thatuh ¼

PNi¼0u

hi , 8uh 2 V h.

Let us consider one substructure at a time. Assume that Xi has vertices

T1; T2; T3 and denote ai ¼ ð1=areaðXiÞÞR

Xiuh dX. Then we have

kIHuhk2L2ðXiÞ 6C kIHuh

� aik2L2ðXiÞ þ kaik2L2ðXiÞ:

Since the function IHuh � ai is linear in the region Xi, a straightforward cal-

culation shows that the L2 norm in Xi can be bounded by

CH 2ððIHuh � aiÞjT1 þ ðIHuh � aiÞjT2 þ ðIHuh � aiÞjT3Þ;

this expression is, using Lemma 2, bounded by Cð1þ lnðH=hÞÞjuhj2H1ðXiÞ, we

bound the kaik2L2ðXiÞ term, by using Schwarz’s inequality

kaik2L2ðXiÞ ¼Z

Xi

1

areaðXiÞ

ZXi

uh dX� �2

dX6 kuhk2L2ðXiÞ;

therefore, we obtain

kIHuhk2L2ðXiÞ 6Cðkuhk2L2ðXiÞ þ H 2ð1þ lnðH=hÞÞjuhj2H1ðXiÞÞ; ð22Þ

using inverse estimate H 2juhj2H1ðXiÞ 6Ckuhk2L2ðXiÞ, we can get

kIHuhk2L2ðXiÞ 6Cð1þ lnðH=hÞÞkuhk2L2ðXiÞ: ð23Þ

By Lemma 1, we have

kuhi k2L2ðX0

iÞ6Ch2

Xxk2Kh\X0

i

ððIhðhiðuh � IHuhÞÞÞðxkÞÞ2:

Since jhij < 1, and xk is a nodal point, the interpolation operator Ih can beremoved. Therefore the right-hand side can be bounded by

Ch2X

xk2Kh\X0i

ððuh � IHuhÞðxkÞÞ2:

Taking into account that the value a nodal point contributes at most three

times, we obtain
Xkuhi k
2

L2ðX0iÞ6Ch2

Xxk2Kh

ððuh � IHuhÞðxkÞÞ2;

we can bound the right-hand side, using Lemma 1, by Ckuh � IHuhk2L2ðXÞ, which

can then be bounded by

Cð1þ lnðH=hÞÞkuhk2L2ðXÞ:


Since IHuh only be decided by the nodal values, so

jIHuhj21;Xi 6Cð1þ lnðH=hÞÞjuhj21;Xi : ð24Þ

Let K denote any of the triangles of the triangulation Th, we can get

juhi j2

1;K 6 2jhixhj21;K þ 2jIhððhi � hiÞxhÞj21;K ; ð25Þ

where hi is the arithmetic mean of hi on K. Using inverse estimate, get

jIhððhi � hiÞxhÞj1;K 6 kIhððhi � hiÞxhÞk0;K ; ð26Þ

and

jhi � hij6Ch=Hi; ðKÞ: ð27Þ

Summing on X0i, we have

juhi j21;X0

i6Cðjxhj1;X0

iþ H�2

i kxhk0;X0iÞ; ð28Þ

but xh ¼ ðI � IH Þuh, use (24) and interpolate estimate, we can get

juhi j2

1;X0i6Cð1þ lnðH=hÞÞkuhk21;Xi

; ð29Þ

the left semi-norm can be replaced by kuhi k21;X0

i, hence

XNi¼0

kuhi k21;X 6Cð1þ lnðH=hÞÞkuhk21;Xi ; ð30Þ

note thatk � ka and k � k1 are equivalent, then

XNi¼0

kuhi k2a6Cð1þ lnðH=hÞÞkuhk2a: ð31Þ

From all of the discussions we have made, we can get

XMi¼0

kuhi k2A ¼

XMi¼0

ðkuhi k2L2ðXÞ þ Dtkuhi k

2aÞ

6Cð1þ lnðH=hÞÞXMi¼0

ðkuhk2L2ðXÞ þ Dtkuhk2aÞ

¼ Cð1þ lnðH=hÞÞkuhk2A:

Thus, we conclude the proof. h


4. Convergence analysis and error estimates

In this section, we give the error estimates of the Scheme 1, that is the proof

of Theorem 2.

Define the auxiliary project uh 2 Vh such that

aðuh � u; vÞ ¼ 0 8v 2 Vh: ð32Þ

Let g ¼ uh � u, then

kgnks þogot

��s

; s ¼ 0; 1: ð33Þ

From (12) we derive that

unh � un�1h

Dt; v

� �þ aðunh; vÞ ¼ ðf ðun�1Þ; vÞ þ ðotgn þ qn; vÞ 8v 2 Vh: ð34Þ

Let en ¼ W n � unh, eni ¼ W n

i � unh, eðj�1ÞM ¼ W ðj�1ÞM � unh, and combining (16)and (34) we have that for any v 2 V h

i ,

enjMþi � en�1

Dt; v

!þ aðenjMþi; vÞ ¼ ðf 0ðen�1 þ gn�1Þ

� otgn � qn; vÞ 8v 2 V h

i ; ð35Þ

enjMþi � ðW ðj�1ÞM � unhÞ x 2 X n X0i; ð36Þ

where f 0 denotes the first order derivative of f at a point between W n�1 and

un�1, f 0ðen�1 þ gn�1Þ ¼ f ðW n�1Þ � f ðun�1Þ. This equation can be changed to

AðenjMþi; vÞ ¼ ðenjMþi; vÞ þ DtaðenjMþi; vÞ¼ ðen�1; vÞ þ Dtðf 0ðen�1 þ gn�1Þ; vÞ � Dtðotgn þ qn; vÞ 8v 2 V h

i :

ð37Þ

Let En 2 Mh be the solution of the equation

AðEn; vÞ ¼ ðEn; vÞ þ DtaðEn; vÞ¼ ðen�1; vÞ þ Dtðf 0ðen�1 þ gn�1Þ; vÞ � Dtðotgn þ qn; vÞ 8v 2 V h

i :

ð38Þ

From (37) and (38) we have that

AðenjMþi � En � ðeðj�1ÞM � EnÞ; vÞ ¼ �Aðeðj�1ÞM � EnÞ; v 8v 2 V h: ð39Þ


Since enjMþi � En � ðeðj�1ÞM � EnÞ 2 V hi , we have

enjMþi � En � ðeðj�1ÞM � EnÞ ¼ �Phi ðeðj�1ÞM � EnÞ;enjMþi � En ¼ ðI � Phi Þðenðj�1ÞM � EnÞ; i ¼ 0; 1; . . . ;M :

ð40Þ

Therefore we have

en � En ¼ ðenmM � EnÞ ¼ 1M

XMi¼0

ðenmMþi � EnÞ

¼ 1M

XMi¼0

ðI � Phi Þðenðm�1ÞM � EnÞ

¼ I

� 1M

XMi¼0Phi

!ðenðm�1ÞM � EnÞ

¼ I

� 1M

XMi¼0Phi

!m

ðen0 � EnÞ: ð41Þ

From Lemmas 3 and 4, we can get

ken � EnkA6 1

�� 1C0

�mken�1 � EnkA;

kenkA6 kEnkA þ C 1

�� 1C0

�mðken�1 � EnkAÞ;

ð42Þ

there C0 ¼ Cð1þ lnðH=hÞÞ.Now we estimate the right-hand side terms of (42). Let v ¼ En in (38), then

we have that

kEnk2A ¼ ðen�1;EnÞ þ Dtðf 0ðen�1 þ gn�1Þ;EnÞ � Dtðotgn þ qn;EnÞ6 ken�1kkEnk þ CDtken�1kkEnk þ CDtðkotgnk þ kqnk þ kgn�1kÞkEnk;

ð43Þ

therefore

kEnkA6 ð1þ CDtÞken�1k þ CDtðkotgnk þ kqnk þ kgn�1kÞ: ð44Þ

From (38) we have that

ðEn � en�1; vÞ þ DtaðEn � en�1; vÞ ¼ Dtðf 0ðen�1 þ gn�1Þ; vÞ � Dtðotgn þ qn; vÞ� Dtaðen�1; vÞ 8v 2 V h: ð45Þ


Let v ¼ En � en�1. Using the same method as (44) we have that

kEn � en�1kA6 ðDtÞ1=2ken�1ka þ CDtðken�1k þ kotgnk þ kqnk þ kgn�1kÞ:ð46Þ

Using (42), (44) and (46) we have that

kenkA6 ð1þ CDtÞken�1k þ C 1

�� 1C0

�mðDtÞ1=2ken�1ka

þ CDt 1

�� 1C0

�mþ kotgnk þ kqnk þ kgn�1k

�: ð47Þ

By 2ab6 a2 þ b2 we have that

ken�1k�

þ 1

�� 1C0

�mðDtÞ1=2ken�1ka

�2

¼ ken�1k2 þ 2C 1

�� 1C0

�mðDtÞ1=2ken�1kaken�1k

þ C2 1�

� 1C0

�2mDtken�1k2a6 ken�1k2 þ Dtken�1k2a

þ C2 1�

� 1C0

�2mken�1k þ C2 1

�� 1C0

�2mDtken�1k2a

6 1

þ C2 1

�� 1C0

�2m!ken�1k2A: ð48Þ

It is clear that

1þ C2 1�

� 1C0

�2m6 1þ 2C 1

�� 1C0

�mþ C2 1

�� 1C0

�2m

¼ 1

�þ C 1

�� 1C0

�m�2;

therefore

1

þ C2 1

�� 1C0

�2m!1=26 1þ C 1

�� 1C0

�m;

so we have

ken�1k þ C 1

�� 1C0

�mðDtÞ1=2ken�1ka6 1

�þ C 1

�� 1C0

�m�ken�1kA:

ð49Þ


Using (48) and (49) we have that

kenkA6 1

�þ C 1

�� 1C0

�m�ken�1kA þ CDt 1

�� 1C0

�mþ kotgnk

þ kqnk þ kgn�1k�; ð50Þ

where the constant C is independent of Dt, h and H .Use the same method as in [6], we can conclude

kW n � unk6 kenk þ kgnk6C Dt�

þ hkþ1 þ 1

�� 1C0

�m�: ð51Þ

Remark 1. Lemmas 3 and 4 and (41) tell us that the convergence rate of

Scheme 1 at each time-level is q ¼ C 1� 1lnðH=hÞ

. h

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additive schwarz methods for parabolic problems

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