9
Two-grid nonconforming finite element method for second order elliptic problems q Cheng Wang a , Ziping Huang b, * , Likang Li c a Department of Applied Mathematics, Tongji University, Shanghai 200092, People’s Republic of China b Chinese-German College, Tongji University, Shanghai 200092, People’s Republic of China c Department of Mathematics, Fudan University, Shanghai 200433, People’s Republic of China Abstract In this paper, we present and analyze two-grid P 1 nonconforming finite element method for second order elliptic prob- lems. The two-grid method involves solving one small origin problem on coarse grid with grid size H, and some correction problems on local fined grids with grid size h. If h = O(H 2 ) is chosen, we show that the convergence rate of this two-grid method is optimal in broken H 1 -norm. Numerical results conforming the theory are also provided. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Second order elliptic problems; P 1 nonconforming finite element; Two-grid method 1. Introduction With the rapid development of parallel computers and parallel algorithms, some extensive interests have focused on the domain decomposition techniques for numerical approximation of partial differential equa- tions. Many domain decomposition algorithms in nature are some forms of generalization or extension away from classical Schwarz method, which is based on decompositions of the domain into overlapping subdo- mains. In most of these algorithms, a large quantity of data should be exchanged and communicated among each subdomain, which leads to low efficiency of algorithm and dependency on computer architecture. Recently, a new domain decomposition technique, termed TGM (two-grid method), has been developed for elliptic problems. The method is a local and parallel method, which also uses a collection of overlapping sub- domains. However, since it is designed upon the local property of the resulting problem, there only need to exchange data once and the quantity of transferring data is also intensively cut down. The two-grid strategy was first introduced by Xu [9,10] for nonsymmetric and nonlinear elliptic problems, and then has been further 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.10.049 q Contract/grant sponsor: National Natural Science Foundation of China; contract/grant number: 40074031. * Corresponding author. E-mail address: [email protected] (Z. Huang). Applied Mathematics and Computation 177 (2006) 211–219 www.elsevier.com/locate/amc

Two-grid nonconforming finite element method for second order elliptic problems

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Page 1: Two-grid nonconforming finite element method for second order elliptic problems

Applied Mathematics and Computation 177 (2006) 211–219

www.elsevier.com/locate/amc

Two-grid nonconforming finite element method for secondorder elliptic problems q

Cheng Wang a, Ziping Huang b,*, Likang Li c

a Department of Applied Mathematics, Tongji University, Shanghai 200092, People’s Republic of Chinab Chinese-German College, Tongji University, Shanghai 200092, People’s Republic of China

c Department of Mathematics, Fudan University, Shanghai 200433, People’s Republic of China

Abstract

In this paper, we present and analyze two-grid P1 nonconforming finite element method for second order elliptic prob-lems. The two-grid method involves solving one small origin problem on coarse grid with grid size H, and some correctionproblems on local fined grids with grid size h. If h = O(H2) is chosen, we show that the convergence rate of this two-gridmethod is optimal in broken H1-norm. Numerical results conforming the theory are also provided.� 2005 Elsevier Inc. All rights reserved.

Keywords: Second order elliptic problems; P1 nonconforming finite element; Two-grid method

1. Introduction

With the rapid development of parallel computers and parallel algorithms, some extensive interests havefocused on the domain decomposition techniques for numerical approximation of partial differential equa-tions. Many domain decomposition algorithms in nature are some forms of generalization or extension awayfrom classical Schwarz method, which is based on decompositions of the domain into overlapping subdo-mains. In most of these algorithms, a large quantity of data should be exchanged and communicated amongeach subdomain, which leads to low efficiency of algorithm and dependency on computer architecture.

Recently, a new domain decomposition technique, termed TGM (two-grid method), has been developed forelliptic problems. The method is a local and parallel method, which also uses a collection of overlapping sub-domains. However, since it is designed upon the local property of the resulting problem, there only need toexchange data once and the quantity of transferring data is also intensively cut down. The two-grid strategywas first introduced by Xu [9,10] for nonsymmetric and nonlinear elliptic problems, and then has been further

0096-3003/$ - see front matter � 2005 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2005.10.049

q Contract/grant sponsor: National Natural Science Foundation of China; contract/grant number: 40074031.* Corresponding author.

E-mail address: [email protected] (Z. Huang).

Page 2: Two-grid nonconforming finite element method for second order elliptic problems

212 C. Wang et al. / Applied Mathematics and Computation 177 (2006) 211–219

investigated by many other researchers [1,4,5,7,8]. Based on the interior estimate skill, Xu [11] extended thetwo-grid method to second order elliptic problems, and proposed a number of local and parallel finite elementalgorithms. The main idea is that first we approximate the elliptic problem on a coarse gird, then the residue,which is dominated by high frequencies, can be corrected by some local and parallel procedures on a collectionof overlapping subdomains. They used standard conforming finite element approximation on both globalcoarse grid and several local fine grids, and obtained an optimal order convergence rate in broken H1-norm.

In this paper, we extend the two-grid finite element method for solving second order elliptic problems to P1

nonconforming finite element case. We use the same P1 nonconforming finite element spaces as coarse spaceand fine space. Because of the lack of nestess of the coarse space and fine space, a suitable intergrid operatormust be chosen. Noting that the two-grid algorithm is carried out on some local fine grids, we construct a spe-cial intergrid operator satisfying an orthogonality on each edge of coarse grid. Based on the properties of thisoperator, we shall show that the optimal order error estimate for the two-grid P1 nonconforming finite elementmethod is also valid.

The remainder of this paper is organized as follows. In Section 2, we describe some fundamental notations,and introduce some useful spaces and broken norms. In Section 3, we propose a local and parallel P1 noncon-forming finite element algorithm. In Section 4, some auxiliary lemmas will be provided. After that, in Section5, we shall demonstrate our main theorem, which indicate the convergence rate of the algorithm. Finally, somenumerical experiments are presented to support the theory.

2. Some notations

In this section, we provide some preliminaries and notations. First we introduce the so-called two-gridpartition, which can be constructed as follows: (1) Give an initial coarse grid TH(X), let us divide X into anumber of nonoverlapping polygonal subdomains Dj such that X ¼

PMj¼1Di, then enlarge each Dj to obtain

a convex polygonal subdomain Xj. In order to obtain the nested local fine grid, we require that Dj and Xj

are composed of elements of coarse grid TH(X). (2) By refining TH(X), we get a global fined grid Th(X) andsome locally refined grids Th(Xj) by restricting Th(X) to Xj. In this paper, we assume that Th(Xj) and TH(X)are two quasi-uniform triangulations [2], with a characteristic grid size h and H, respectively.

Associated with these grids, we introduce some finite element spaces. The P1 conforming finite elementspaces are defined to be

V hðXÞ ¼ fv: v is continuous in X and vj ¼ vjKjis linear for all Kj 2 T hðXÞg;

V h0ðXÞ ¼ fv: v 2 V hðXÞ and v ¼ 0 at the vertices on oXg.

The P1 nonconforming finite element spaces are defined to be

ShðXÞ ¼ fv: vj ¼ vjKjis linear for all Kj 2 T hðXÞ;

v is continuous at the midpoints of the edgesg;Sh

0ðXÞ ¼ fv: v 2 ShðXÞ and v ¼ 0 at the midpoints of edges on oXg;

and let Sh(X0) and Th(X0) be the restriction of Sh(X) and Th(X) to X0 � X. For any X0 or D mentioned in thispaper, we assume that it is one of the subdomains Xj or Dj (j = 1, . . . ,M). Moreover, we define

S0hðX0Þ ¼ fv : v 2 ShðXÞ and supp v �� X0g.

Here and thereafter, for G � X0, we use G �� X0 to mean that dist(oGnoX,oX0noX) > 0 (see [11] for detail).The so-called broken norm and semi-norm are also defined

jvjm;h;X ¼X

K2T hðXÞjvj2m;K

!1=2

; kvkm;h;X ¼X

K2T hðXÞkvk2

m;K

!1=2

; m ¼ 0; 1; 2.

In this paper, we will adopt the standard definitions for Sobolev spaces and their norms and semi-norms [3],and denote by C or c a generic positive constant independent of related parameters (in particular of grid size hand H), which may stand for different values in different places.

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C. Wang et al. / Applied Mathematics and Computation 177 (2006) 211–219 213

3. Two-grid nonconforming finite element method

For simplicity, let us consider the following model problem:

�Du ¼ f in X;

u ¼ 0 on C;ð3:1Þ

where X is a plane convex polygonal domain with the boundary C and f 2 L2(X). The weak formulation of(3.1) is to find u 2 H 1

0ðXÞ such that

aXðu; vÞ ¼ ðf ; vÞX 8v 2 H 10ðXÞ; ð3:2Þ

where, and throughout this paper, for any domain V � R2, (Æ, Æ)V is an inner product of L2(V) and

aV ðw; vÞ ¼Z

Vrwrvdx. ð3:3Þ

It is well known that the problem (3.1) admits a unique solution in the class H 2ðXÞ \ H 10ðXÞ.

The P1 nonconforming finite element discretization for the problem (3.2) can be written as: Find u 2 Sh0ðXÞ

such that

ahXðuh; vÞ ¼ ðf ; vÞX 8v 2 Sh

0ðXÞ; ð3:4Þ

where

ahXðw; vÞ ¼

XK2T hðXÞ

aKðw; vÞ. ð3:5Þ

We now construct the two-grid method for P1 nonconforming finite element. Since SH0 ðXÞ 6� Sh

0ðXÞ, we mustchoose a suitable coarse-to-fine intergrid operator I.

For v 2 SH0 ðXÞ, we define the intergrid operator I : SH

0 ðXÞ ! V H=20 ðXÞ as follows:

(a) If p is a vertex of TH(X) along oX, IvðpÞ ¼ 0.(b) If p is a interior vertex of TH(X), v may have a jump at p and IvðpÞ takes the average of v at p.(c) If m is a midpoint of edge p1p2;IvðmÞ ¼ 2vðmÞ � 1

2Ivðp1Þ � 1

2Ivðp2Þ.

Based on the above intergrid operator, we describe two-grid P1 nonconforming finite element method infollowing three steps, which will also be named as TGM.

Step 1. Find a global coarse grid solution uH 2 SH0 ðXÞ:

aHX ðuH ; vÞ ¼ ðf ; vÞX 8v 2 SH

0 ðXÞ.

Step 2. Find local fine grid correction ej

h 2 Sh0ðXjÞ (j = 1, . . . ,M) in parallel:

ahXjðej

h; vÞ ¼ ðf ; vÞXj� ah

XjðIuH ; vÞ 8v 2 Sh

0ðXjÞ.

Step 3. Set uh ¼ IuH þ ejh, in Dj (j = 1, . . . ,M).

4. Local behavior of P1 nonconforming finite element

In this section, we prove an local a prior estimate of P1 nonconforming finite element approximation, whichplays an important role in the later convergence analysis of the two-grid method. Firstly, we present some lem-mas on the P1 nonconforming finite element space.

Lemma 4.1 [2]. For any v 2 Sh(X), we have

kvk1;h;X 6 Ckh�1vk0;X. ð4:6Þ

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214 C. Wang et al. / Applied Mathematics and Computation 177 (2006) 211–219

Lemma 4.2 [2]. For any w 2 H 2ðX0Þ \ H 10ðX0Þ, there exists v 2 Sh

0ðX0Þ such that

kw� vk0;X0þ hkw� vk1;h;X0

þ h2kw� vk2;h;X06 Ch2kwk2;X0

. ð4:7Þ

Lemma 4.3. For D � X0 �� X, let x 2 C10 ðXÞ with suppx �� D. Then, for each w 2 Sh(D), there exists

v 2 S0hðDÞ such that

jjxw� vjj1;h;D 6 Chjjwjj1;h;D. ð4:8Þ

Proof. Let v be the interpolation of xw in Sh0ðDÞ. Since suppx �� D, v 2 S0

hðDÞ. Noticing that w is linear ineach element K 2 Th(D), the result (4.8) then follows from the approximation properties, that is,

kxw� vk1;K 6 ChK jxwj2;K 6 ChKkwk1;K . �

For the P1 nonconforming finite element solution of problem (3.4), it is well known that we have the fol-lowing estimates.

Theorem 4.1 [2,6]. Assume that u and uh are the solutions of (3.2) and (3.4), respectively. Then the error satisfies

the estimates

ku� uhk1;h;X 6 Chkuk2;X; ð4:9Þku� uhk0;X 6 Ch2kuk2;X. ð4:10Þ

Lemma 4.4. Assume that X0 �� X and x 2 C10 ðXÞ satisfying suppx �� X0, then we have

ahXðxw;xwÞ 6 ah

Xðw;x2wÞ þ Cjjwjj20;X08w 2 Sh

0ðXÞ. ð4:11Þ

Proof. For any w 2 Sh0ðXÞ, we have

ahXðxw;xwÞ ¼ ah

Xðw;x2wÞ þX

K2T hðXÞ

ZK

oxox1

oxox1

þ oxox2

oxox2

� �w2

� �dx 6 ah

Xðw;x2wÞ þ Cjjwjj20;X0. �

We now give a result similar to Lemma (3.2) in [11], which describe the local behavior of P1 nonconformingfinite element approximation.

Lemma 4.5. Assume that f 2 L2(X) and w 2 Sh(X0) satisfy

ahX0ðw; vÞ ¼ ðf ; vÞX0

8v 2 S0hðX0Þ. ð4:12Þ

Then, if D �� X0 �� X, we have

jjwjj1;h;D 6 Cðjjwjj0;X0þ jjf jj0;X0

Þ. ð4:13Þ

Proof. Let X1, X2 satisfy

D �� X2 �� X1 �� X0.

Choose D1 � X satisfying D �� D1 �� X2 and x 2 C10 ðXÞ such that x � 1 on D1 and suppx �� X2. Thenfrom Lemma 4.3, there exists v 2 S0

hðX2Þ such that

jjx2w� vjj1;h;X26 Chjjwjj1;h;X2

; ð4:14Þ

which implies

ahX0ðw;x2w� vÞ 6 Chjjwjj21;h;X2

ð4:15Þ

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C. Wang et al. / Applied Mathematics and Computation 177 (2006) 211–219 215

and

jðf ; vÞX0j 6 Cjjf jj0;X0

jjvjj0;X26 Cjjf jj0;X0

ðhjjwjj1;h;X2þ jjxwjj1;h;X2

Þ. ð4:16Þ

Since v 2 S0hðX2Þ � S0

hðX0Þ, we get

ahX0ðw;x2wÞ ¼ ah

X0ðw;x2w� vÞ þ ah

X0ðw; vÞ ¼ ah

X0ðw;x2w� vÞ þ ðf ; vÞX0

. ð4:17Þ

Thus, from (4.14)–(4.17), and Lemma 4.4 we have

kxwk21;h;X0

6 CðahX0ðxw;xwÞ þ kwk2

0;X0Þ 6 Cðah

X0ðw;x2wÞ þ kwk2

0;X0Þ

6 Cðhkwk21;h;X2

þ kf k0;X0ðhkwk1;h;X2

þ kxwk1;h;X2Þ þ kwk2

0;X0Þ.

An application of e-inequality yields

kwk1;h;D 6 Cðh1=2kwk1;h;X2þ kwk0;X0

þ kf k0;X0Þ.

Similar argument leads to

kwk1;h;X26 Cðh1=2kwk1;h;X0

þ kwk0;X0þ kf k0;X0

Þ.

Combining the above two estimates with Lemma 4.1, we obtain

jjwjj1;h;D 6 Cðhjjwjj1;h;X0þ jjwjj0;X0

þ jjf jj0;X0Þ 6 Cðjjwjj0;X0

þ jjf jj0;X0Þ. �

5. Convergence analysis

In this section, we aim to derive error estimates for two-grid P1 nonconforming finite element method.Firstly, we provide some lemmas about the intergrid operator I.

The following lemma is a direct result of scaling argument.

Lemma 5.1. For any v 2 Sh(X), K 2 Th(X), we have

ckvk20;K 6 h2

K

X3

i¼1

v2ðmiÞ 6 Ckvk20;K ; ð5:18Þ

where mi is the midpoint of edge of K.

From Lemma 5.1 and the definition of I, it is easy to show that

Lemma 5.2. For the intergrid operator I, we have

kIvk0;X 6 Ckvk0;X 8v 2 SH0 ðXÞ; ð5:19Þ

Iv ¼ v 8v 2 V H0 ðXÞ; ð5:20ÞZ

eðIv� vÞds ¼ 0 8v 2 SH

0 ðXÞ; ð5:21Þ

where e is an edge of grid TH(X).

Now we are ready to prove an error estimate of TGM on a subdomain D �� X.

Theorem 5.1. Assume that uh 2 Sh(X0) is obtained by TGM, and uh is a global P1 nonconforming element

solution of (3.4) on the grid Th(X). If h = O(H2), then

jjuh � uhjj1;h;D 6 Chjjujj2;X. ð5:22Þ

Proof. By the definition of TGM, we have

ahX0ðuh � uh; vÞ ¼ 0 8v 2 S0

hðX0Þ. ð5:23Þ

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216 C. Wang et al. / Applied Mathematics and Computation 177 (2006) 211–219

Then, Lemma 4.5 and (5.23) imply that

jjuh � uhjj1;h;D 6 Cjjuh � uhjj0;X06 Cðjjuh �IuH jj0;X0

þ jjehjj0;X0Þ. ð5:24Þ

From (5.19) and (5.20), we get

jjuh �IuH jj0;X06 jjuh �IuH jj0;X0

þ jjIuH �IuH jj0;X06 jjuh � uH jj0;X0

þ jjIðuH � uH Þjj0;X06 CH 2kuk2;X;

ð5:25Þ

where uH 2 V H ðXÞ is P1 conforming finite element approximation of (3.2) on TH(X).

To estimate jjehjj0;X0, we employ the Aubin–Nitsche duality argument. Consider the dual problem:

�Dw ¼ eh; x 2 X0;

w ¼ 0; x 2 oX0.ð5:26Þ

Since X0 is convex, we assume that it has a unique solution and the following regularity estimate holds:

kwk2;X 6 Ckehk0;X. ð5:27Þ

Let n 2 V H0 ðX0Þ be the P1 conforming finite element approximation of (5.26) on TH(X0), then we have

kehk20;X0¼ ð�Dw; ehÞX0

¼ ahX0ðw; ehÞ �

XK2T hðX0Þ

ZoK

owonK

eh ds

¼ ahX0ðeh;w� nÞ þ ah

X0ðeh; nÞ �

XK2T hðX0Þ

ZoK

owonK

eh ds; ð5:28Þ

where ni denotes the unit outward normal related to K.Now, let us bound each of the terms on the right-hand side of (5.28).From the approximation estimates of n and (5.27), we get

jahX0ðeh;w� nÞj 6 Ckehk1;h;X0

kw� nk1;X06 CHkehk1;h;X0

kwk2;X06 CHkehk1;h;X0

kehk0;X0. ð5:29Þ

On the other hand, application of Lemma 4.2 and Cauchy–Schwartz inequality gives

XK2T hðX0Þ

ZoK

owonK

eh ds

���������� ¼

XK2T hðX0Þ

ZoK

oðw� vKÞonK

ðeh � ehÞds

����������

6 CX

K2T hðX0Þkw� vKk1;oKkeh � ehk0;oK

6 CHX

K2T hðX0Þh�1kw� vKk2

1;K þ hkw� vKk22;K

!1=2

kehk1;h;X0

6 CH 2kehk0;X0kehk1;h;X0

; ð5:30Þ

where eh is the average of eh on each edge of element K and v 2 Sh

0ðX0Þ satisfying Lemma 4.2.What left is to estimate the second term of (5.28). However, notice that n is linear in each element K of

coarse grid TH(X0), and IuH is continuous in X0. Then, it follows from Lemma 5.2 that

ahX0ðeh; nÞ ¼ aH

X0ðuH ; nÞ � aX0

ðIuH ; nÞ

¼X

K2T H ðX0Þ

ZKrðuH Þrndx�

XKi2T H=2ðX0Þ

ZKrðIuH Þrndx

¼X

K2T H ðX0Þ

ZoK

uHonK

onKdx�

XK2T H=2ðX0Þ

ZoK

IuHonK

onKdx ¼

XK2T H ðX0Þ

ZoKðuH �IuH Þ

onK

onKdx ¼ 0:

ð5:31Þ

Using the similar method as in the deduction of (5.25), we have

kehk21;h;X 6 Cah

X0ðeh; ehÞ 6 Cah

X0ðuh �IuH ; ehÞ 6 CHkuk2;Xkehk1;h;X. ð5:32Þ

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C. Wang et al. / Applied Mathematics and Computation 177 (2006) 211–219 217

Combining the above inequalities, we obtain

kehk0;X 6 CH 2kuk2;X.

This together with (5.24) and (5.25) completes proof. h

Observing that the above estimate only holds on one subdomain D, we need the following broken norm toobtain the global error estimate of TGM. Define

jjjuh � uhjjj :¼XM

i¼1

jjuh � uhjj1;h;Di. ð5:33Þ

The main theorem of this paper then is a direct consequence of the above broken norm definition andTheorem 5.1.

Theorem 5.2. Assume that uh is obtained by TGM, and uh is a global P1 nonconforming element solution of (3.4)

on the grid Th(X). If h = O(H2), then

jjjuh � uhjjj 6 Chjjujj2;X. ð5:34Þ

6. Numerical experiments

In this section, some numerical experiments will be carried out to check our theory.We consider the simple unit square domain X = (0, 1) · (0, 1) and a uniform triangulation Th(X) = {K} (see

Fig. 1).

Fig. 1. Triangulation.

Fig. 2. Domain decomposition.

Page 8: Two-grid nonconforming finite element method for second order elliptic problems

Table 1Results of Example 1

h kjuh � ukj1,Xjjju4h�ujjj1;Xjjjuh�ujjj1;X

2�6 3.627E�032�8 8.930E�04 4.062�10 2.224E�04 4.02

Table 2Results of Example 2

h kjuh � ukj1,Xjjju4h�ujjj1;Xjjjuh�ujjj1;X

2�6 1.351E�022�8 3.412E�03 3.962�10 8.526E�04 4.00

218 C. Wang et al. / Applied Mathematics and Computation 177 (2006) 211–219

Subdomains Di and Xi are set as follows (Fig. 2):

D1 ¼ ð0; 1=2Þ � ð0; 1=2Þ; D2 ¼ ð0; 1=2Þ � ð1=2; 1Þ;D3 ¼ ð1=2; 1Þ � ð0; 1=2Þ; D4 ¼ ð1=2; 1Þ � ð1=2; 1Þ;X1 ¼ ð0; 5=8Þ � ð0; 5=8Þ; X2 ¼ ð0; 5=8Þ � ð3=8; 1Þ;X3 ¼ ð3=8; 1Þ � ð0; 5=8Þ; X4 ¼ ð3=8; 1Þ � ð3=8; 1Þ.

We consider the following two examples:

Example 1

�Du ¼ 2ðð1� yÞy þ ð1� xÞxÞ in X;

u ¼ 0 on oX.

Example 2

�Du ¼ sinðpxÞðp2ð1� yÞy7 þ 56y6 � 42y5Þ in X;

u ¼ 0 on oX.

We shall apply TGM to solve these problems with fine grids of size h = 2�j (j = 6,8,10) and the correspond-ing coarse grids of size H = h1/2. The numerical results of Examples 1 and 2 are listed in Tables 1 and 2, fromwhich we observe that the convergence rate of uh is O(h) as h tends to zero. This well coincides with our the-oretic analysis in this paper.

References

[1] O. Axelsson, W. Layton, A two-level discretization of nonlinear boundary value problems, SIAM J. Numer. Anal. 33 (1996) 2359–2374.

[2] S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, Berlin, Heidelberg, 1994.[3] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.[4] C.N. Dawson, M.F. Wheeler, Two-grid methods for mixed finite element approximations of nonlinear parabolic equations, Contemp.

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J. Numer. Anal. 35 (1998) 435–452.[6] J. Douglas Jr., J.E. Santos, D. Sheen, X. Ye, Nonconforming Galerkin methods based on quadrilateral elements for second order

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[8] C. Wang, Z.P. Huang, L.K. Li, Two-grid covolume schemes for elliptic problems, in press.[9] J.C. Xu, A new class of iterative methods for nonselfadjoint or indefinite problems, SIAM J. Numer. Anal. 29 (1992) 303–319.

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Further reading

[1] M. Crouzeix, P.A. Raviart, Conforming and non-conforming finite element methods for solving the stationary Stokes equations,RAIRO Anal. Numer. 7 (1996) 33–76.