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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).
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.
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
whereahXðw; 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 ejh 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
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
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
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 hðX0ÞZoK
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 Sh0ðX0Þ 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
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.
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
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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.