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Superconvergence of H(div) finite element approximations for the Stokes problem by L2 -projection methods
Applied Mathematics and Computation 219 (2013) 5649–5656
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Superconvergence of Hðdiv Þ finite element approximations for the Stokes problem by L2 -projection methods Rabeea Jari a,⇑, Lin Mu a, Xiu Ye b a b
Department of Applied Science, University of Arkansas at Little Rock, 2801 South University, Little Rock, AR 72204, United States Department of Mathematics, University of Arkansas at Little Rock, 2801 South University, Little Rock, AR 72204, United States
a r t i c l e
i n f o
Keywords: Finite element methods Superconvergence L2 -projection The Stokes equations
a b s t r a c t A general superconvergence result of Hðdiv Þ finite element approximations for the Stokes equations is established by using L2 -projection method. Regularity assumptions for the Stokes problem with regular partitions is required. Numerical experiments are given to verify the theoretical results. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction The superconvergence of finite element solutions is an interesting and useful phenomenon in scientific computing of real world problems. In recent years, the superconvergence for finite element solutions has been an active research area in numerical analysis [6,7,11,12,16–18,22–26]. The purpose of this paper is to improve the existing approximation accuracy by applying certain postprocessing techniques which is easy to implement with little cost. L2 -projection method was proposed and analyzed to gain superconvergence by Wang [20] for the second order elliptic problem. It has been extended by Wang and Ye [21] to the Stokes equations for H1 conforming finite element method [8,9,13]. This paper aims at a study of superconvergence by using the L2 -projection method to the Hðdiv Þ finite element for the Stokes equations. The superconvergence result is based on some regularity assumption of the exact solution for the Stokes problem. The continuity equation r uh ðxÞ ¼ 0 8x 2 X requires the numerical solution uh to be a member of the Sobolev space Hðdiv Þ. To satisfy this property, several finite element schemes have been developed to generate locally divergence-free solutions [4]. Particularly, a recent approach by using Hðdiv Þ conforming finite elements method has been proposed and studied for incompressible fluid flow problems [27]. The main advantage of Hðdiv Þ finite element methods is that the discrete velocity field is exactly divergence-free. Hence, it seems that the Hðdiv Þ finite element might be easily decoupled between the velocity and the pressure unknowns, since this methods satisfy the continuity equation and the inf–sup condition of Brezzi [2] and Babuška [1]. The remainder of the paper is organized as follows: in Section 2, the Stokes equations is presented, along with variational formulation of Hðdiv Þ finite element for the Stokes equations and its finite element scheme. In Section 3, we establish superconvergence error estimates for both velocity and pressure approximations of Hðdiv Þ finite element for the Stokes problem by using L2 -projection method. Finally, in Section 4, several numerical experiments are present to support the theoretical conclusion.
⇑ Corresponding author. E-mail addresses: [email protected] (R. Jari), [email protected] (L. Mu), [email protected] (X. Ye). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.11.023
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2. Preliminaries for the Stokes problem Let us consider the Stokes equations with the homogeneous boundary condition for the velocity variable. Let X be a bounded, connected, and open subset of R2 . The Stokes problem seeks the velocity u and the pressure p in a certain functional spaces such that
mDu þ rp ¼ f in X; r u ¼ 0 in X; u ¼ 0 on @ X;
ð2:1Þ ð2:2Þ ð2:3Þ
where m denotes the fluid viscosity; D; r, and r denote the Laplacian, gradient, and divergence operators, respectively; f ¼ fðxÞ 2 ðL2 ðXÞÞ2 is the unit external volumetric force acting on the fluid at x 2 X [28,30]. In this paper, we assume that m ¼ 1. This paper will use the standard notations for the Sobolev spaces Hs ðXÞ and their associated inner products ð; Þs , norms k ks , and semi-norms j js for s P 0. The Sobolev space H0 ðXÞ coincides with the space of square integrable functions L2 ðXÞ, in which case the norm and inner product are denoted by k k and ð; Þ, respectively. Moreover, denote by L20 ðXÞ the subspace of L2 ðXÞ consisting of the functions in L2 ðXÞ vanishing mean value. The space Hðdiv Þ is a set of vector-valued functions on X defined as follows,
Hðdiv Þ ¼ fv : v 2 ðL2 ðXÞÞ2 ; r v 2 L2 ðXÞg equipped with the norm 1
kv kHðdiv Þ ¼ ðkv k2 þ kr v k2 Þ2 : Let T h be a finite element partition of the domain X with characteristic mesh size h, assume that the partition T h is quasiuniform: i.e, it is shape regular and satisfies the inverse assumption [3]. Define finite element spaces V h and W h for the velocity and pressure, respectively, by
V h ¼ fv 2 Hðdiv Þ : v jK 2 V k ðKÞ;
8K 2 T h ;
W h ¼ fq 2 L20 ðXÞ : qjK 2 W m ðKÞ;
8K 2 T h g;
v nj@X ¼ 0g;
where n is the outward normal direction on the boundary of X; V k ðKÞ is a space of vector-valued polynomials on the element K with index k P 1, and W m ðKÞ is a space of polynomials on the element K with index m P 0. Let e be an interior edge shared by two elements K 1 and K 2 , and let n1 and n2 be unit normal vectors on e pointing exterior to K 1 and K 2 , respectively. Denote by s1 and s2 the two unit tangential vectors which make the right-hand coordinate systems with n1 and n2 , respectively. We define the average fg and jump ½ on e for vector-valued functions w as follows:
feðwÞg ¼
1 ðn1 rðw s1 Þj@K 1 þ n2 rðw s2 Þj@K 2 Þ 2
and
½w ¼ wj@K 1 s1 þ wj@K 2 s2 : If e is an edge on the boundary X with e 2 @K 1 \ @K 2 , the above two operations modified by
feðwÞg ¼ n1 rðw s1 Þj@K 1 ; ½w ¼ wj@K 1 s1 :
v
Let E h denote the union of the boundaries of all elements K 2 T h . Let VðhÞ ¼ V h þ ðHs ðXÞ \ H10 ðXÞÞ2 , with s > 32. For 2 VðhÞ, define rh v to be the function whose restriction to each element K 2 T h is given by the standard gradient rv . The Hðdiv Þ finite element approximation problem of (2.1)–(2.3) seeks ðuh ; ph Þ 2 V h W h such that
aðuh ; v Þ bðv ; ph Þ ¼ ðf; v Þ; bðuh ; qÞ ¼ 0;
8v 2 V h ;
ð2:4Þ
8q 2 W h ;
ð2:5Þ
with
aðuh ; v Þ ¼ ðrh uh ; rh v Þ
XZ e2eh
and
bðuh ; qÞ ¼ ðr uh ; qÞ;
e
feðuh Þg½v ds þ
XZ e2eh
e
1
ðdhe ½uh ½v feðv Þg½uh Þds
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where d is a positive parameter that should be large enough to guarantee a good convergence rate and as small as possible in order to keep the condition number of the discrete system low, (see [29]), and he is the length of the edge e. The exact solution ðu; pÞ of the Stokes problem belongs to VðhÞ satisfies the following variational equations:
aðu; v Þ bðv ; pÞ ¼ ðf; v Þ; bðu; qÞ ¼ 0;
8 v 2 V h;
ð2:6Þ
8 q 2 W h:
ð2:7Þ
We introduce two norms jjj jjj and jjj jjj1 for the set VðhÞ as follows:
jjjv jjj21 ¼ jv j21;h þ
X 1 he k½v k2e ;
ð2:8Þ
e2eh
jjjv jjj2 ¼ jjjv jjj21 þ
X
he kfeðv Þgk2e ;
ð2:9Þ
e2eh
where
jv j21;h ¼
X
jv j21;K and kv k2e ¼
K2T h
Z
v v ds:
e
The following lemma and theorem can be found in [27]. Lemma 2.1. There exists a constant C independent of h such that
jaðw; v Þj 6 Cjjjwjjj jjjv jjj; jbðw; qÞj 6 Cjjjwjjj kqk;
8w;
v 2 VðhÞ;
8w 2 VðhÞ;
q 2 L20 ðXÞ:
Theorem 2.1. Let ðu; pÞ be the solution of (2.1)–(2.3) and ðuh ; ph Þ 2 V h W h be the solution of (2.4) and (2.5), there exists a constant C independent of h such that k
jjju uh jjj þ kp ph k 6 Ch ðkukkþ1 þ kpkk Þ:
ð2:10Þ
In the next section, the superconvergence analysis requires certain regularity for the Stokes problem. To this end, we consider a more general Stokes problem: find ðu; pÞ 2 ðH10 ðXÞÞ2 L20 ðXÞ satisfying
8v 2 ðH10 ðXÞÞ2 ;
aðu; v Þ bðv ; pÞ ¼ ðf; v Þ; bðu; qÞ ¼ ðg; qÞ;
8q 2
ð2:11Þ
L20 ð
XÞ;
L20 ð
ð2:12Þ s
where g 2 XÞ is a given function. Assume that the domain X is so regular that ensures a H ; s P 1 regularity for the solution of (2.11) and (2.12). In other words, for any f 2 ðHs2 ðXÞÞ2 and g 2 Hs1 ðXÞ \ L20 ðXÞ, the problem (2.11) and (2.12) has a unique solution u 2 ðH10 ðXÞÞ2 \ ðHs ðXÞÞ2 and p 2 Hs1 ðXÞ \ L20 ðXÞ satisfying the following a priori estimate:
kuks þ kpks1 6 Cðkfks2 þ kgks1 Þ;
ð2:13Þ
where C is a constant independent of the data f and g, [21]. 3. Superconvergence by L2 -projection The L2 -projection technique was introduced by Wang [20]. It projects the approximate solution to another finite element dimensional space associated with a coarse mesh. The difference of the size in the two mesh sizes can be used to achieve a superconvergence. In order to extend the idea in [20] to the Hðdiv Þ finite element approximation for the Stokes problem, we start with defining a coarse mesh T s where s h satisfying:
s ¼ ha ;
ð3:1Þ
with a 2 ð0; 1Þ. For now, the parameter a will play an important role in the postprocessing. Define two finite element spaces V s ðHs2 ðXÞÞ2 and W s Hs1 ðXÞ for the velocity and pressure, respectively. Here V s and W s are consisting of piecewise polynomials of degree r and t, respectively associated with the partition T s . Let Q s and Rs to be the L2 -projectors from L2 ðXÞ onto the finite element spaces V s and W s , respectively. In the following, we will analyze the error of u Q s uh and p Rs ph . The following lemma provides an error estimate for Q s u Q s uh . Lemma 3.1. Suppose that (2.13) holds with 1 6 s 6 k þ 1 and V s ðHs2 ðXÞÞ2 . Then there is a constant C independent of h and s such that
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kQ s u Q s uh k 6 Ch
s1þa minð0;2sÞ
ðjjju uh jjj þ kp ph kÞ;
ð3:2Þ
where a 2 ð0; 1Þ is as defined in (3.1). Proof. Using the definition of k k and Q s , we have
kQ s u Q s uh k ¼
sup
jðQ s u Q s uh ; /Þj:
/2ðL2 ðXÞÞ2 ;k/k¼1
From the definition of the L2 -projection for Q s , we have
ðQ s u Q s uh ; /Þ ¼ ðu uh ; Q s /Þ: It follows directly that
kQ s u Q s uh k ¼
sup
jðu uh ; Q s /Þj:
ð3:3Þ
/2ðL2 ðXÞÞ2 ;k/k¼1
Consider the following problem: find ðw; kÞ 2 ðH10 ðXÞÞ2 L20 ðXÞ such that
8v 2 ðH10 ðXÞÞ2 ;
aðw; v Þ bðv ; kÞ ¼ ðQ s /; v Þ; bðw; qÞ ¼ 0;
ð3:4Þ
8 q 2 L20 ðXÞ:
ð3:5Þ
The difference of (2.4) and (2.6) gives
aðu uh ; v Þ bðv ; p ph Þ ¼ 0;
8v 2 V h :
ð3:6Þ
The difference of (2.5) and (2.7) yields
bðu uh ; qÞ ¼ 0; By setting
8 q 2 W h:
v in (3.4) by u uh and then using (3.6) and Lemma 2.1, we obtain
ðu uh ; Q s /Þ ¼ aðw; u uh Þ bðu uh ; kÞ ¼ aðw wI ; u uh Þ þ bðwI ; p ph Þ bðu uh ; k kI Þ ¼ aðw wI ; u uh Þ bðw wI ; p ph Þ bðu uh ; k kI Þ 6 Cðjjjw wI jjj jjju uh jjj þ jjjw wI jjj kp ph k þ jjju uh jjj kk kI kÞ;
ð3:7Þ
where wI 2 V h and kI 2 W h are two interpolation functions. Using the Schwarz inequality and (2.10) we have
jðu uh ; Q s /Þj 6 Cðjjjw wI jjj þ kk kI kÞðjjju uh jjj þ kp ph kÞ 6 Ch
s1
ðkwks þ kkks1 Þðjjju uh jjj þ kp ph kÞ:
In addition, from (2.13), the inverse inequality and (3.1) we obtain
jðu uh ; Q s /Þj 6 Ch 6 Ch
s1
kQ s /ks2 ðjjju uh jjj þ kp ph kÞ 6 Ch
s1þa minð0;2sÞ
s1 minð0;2sÞ
s
ðjjju uh jjj þ kp ph kÞk/k
ðjjju uh jjj þ kp ph kÞk/k:
The last estimate, together with (3.3), completes the proof of the lemma. h The following lemma provides an error estimate for Rs p Rs ph . Lemma 3.2. Suppose that (2.13) holds with 1 6 s 6 k þ 1 and W s Hs1 ðXÞ. Then there is a constant C independent of h and s such that
kRs p Rs ph k 6 Ch
s1þa minð0;1sÞ
ðjjju uh jjj þ kp ph kÞ;
ð3:8Þ
where a 2 ð0; 1Þ is as defined in (3.1). Proof. From the definition of k k and Rs , we have
kRs p Rs ph k ¼
sup
jðRs p Rs ph ; /Þj:
/2L20 ðXÞ;k/k¼1
From the definition of the L2 -projection for Rs , we have
ðRs p Rs ph ; /Þ ¼ ðp ph ; Rs /Þ: It follows directly that,
kRs p Rs ph k ¼
sup
jðp ph ; Rs /Þj:
/2L20 ðXÞ;k/k¼1
ð3:9Þ
R. Jari et al. / Applied Mathematics and Computation 219 (2013) 5649–5656
5653
Consider the following problem: find ðx; nÞ 2 ðH10 ðXÞÞ2 L20 ðXÞ such that
8v 2 ðH10 ðXÞÞ2 ;
aðx; v Þ bðv ; nÞ ¼ 0; bðx; qÞ ¼ ðRs /; qÞ;
8q 2
L20 ð
ð3:10Þ
XÞ:
ð3:11Þ
Setting q in (3.11) by p ph and using (3.6), we obtain
ðp ph ; Rs /Þ ¼ bðx; p ph Þ ¼ bðx xI ; p ph Þ þ bðxI ; p ph Þ ¼ bðx xI ; p ph Þ þ aðu uh ; xI Þ ¼ bðx xI ; p ph Þ þ aðu uh ; xI xÞ þ aðu uh ; xÞ ¼ bðx xI ; p ph Þ þ aðu uh ; xI xÞ þ bðu uh ; nÞ ¼ bðx xI ; p ph Þ þ aðu uh ; xI xÞ þ bðu uh ; n nI Þ; where xI 2 V h and nI 2 W h are two interpolation functions. Using the Schwarz inequality we obtain
jðp ph ; Rs /Þj 6 C ðjjjx xI jjjkp ph k þ jjju uh jjjjjjx xI jjj þ kn nI kjjju uh jjjÞ 6 Cðjjjx xI jjj þ kn nI kÞðjjju uh jjj þ kp ph kÞ: Using (2.10) then,
jðp ph ; Rs /Þj 6 Ch
s1
ðkxks þ knks1 Þðjjju uh jjj þ kp ph kÞ:
Furthermore, from (2.13) we have,
jðp ph ; Rs /Þj 6 Ch 6 Ch
s1
kRs /ks1 ðjjju uh jjj þ kp ph kÞ 6 Ch
s1þa minð0;1sÞ
s1 minð0;1sÞ
s
ðjjju uh jjj þ kp ph kÞk/k
ðjjju uh jjj þ kp ph kÞk/k:
The last estimate, together with (3.9), completes the proof of the lemma. Now, we can estimate u Q s uh .
h
Theorem 3.1. Assume that (2.13) holds true with 1 6 s 6 k þ 1 and V s ðHs2 ðXÞÞ2 . If ðuh ; ph Þ is the finite element approximation of the solution ðu; pÞ, then we have a
ku Q s uh k þ h krs ðu Q s uh Þk 6 Ch where
aðrþ1Þ
r
kukrþ1 þ Ch ðjjju uh jjj þ kp ph kÞ;
ð3:12Þ
r ¼ s 1 þ a minð0; 2 sÞ.
Proof. Since Q s is the L2 -projection, then
ku Q s uk 6 C srþ1 kukrþ1 6 Ch
aðrþ1Þ
kukrþ1 :
ð3:13Þ
Combining (3.13) and (3.2) gives
ku Q s uh k 6 ku Q s uk þ kQ s u Q s uh k 6 Ch
aðrþ1Þ
kukrþ1 þ Ch
s1þa minð0;2sÞ
ðjjju uh jjj þ kp ph kÞ;
which completes the estimate for ku Q s uh k in (3.12). Next, we estimate krs ðu Q s uh Þk as follows. Since Q s is the L2 -projection, then ar
krs ðu Q s uÞk 6 C sr kukrþ1 6 Ch kukrþ1 :
ð3:14Þ
a
Multiplying both sides of (3.14) by h , we get a
aðrþ1Þ
h krs ðu Q s uÞk 6 Ch
kukrþ1 :
ð3:15Þ
Moreover, the inverse inequality gives
krs ðQ s u Q s uh Þk 6 C s1 kQ s u Q s uh k:
ð3:16Þ
Using (3.1), the above inequality implies a
h krs ðQ s u Q s uh Þk 6 CkQ s u Q s uh k:
ð3:17Þ
Consequently, from (3.2), we obtain a
h krs ðQ s u Q s uh Þk 6 Ch
s1þa minð0;2sÞ
ðjjju uh jjj þ kp ph kÞ:
ð3:18Þ
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Combining (3.15) and (3.18) gives a
a
a
h krs ðu Q s uh Þk 6 h krs ðu Q s uÞk þ h krs ðQ s u Q s uh Þk 6 Ch
aðrþ1Þ
kukrþ1 þ Ch
s1þa minð0;2sÞ
ðjjju uh jjj þ kp ph kÞ:
This completes the proof of the theorem. h Likewise, we can derive a similar result for the pressure. Theorem 3.2. Suppose that (2.13) holds true with 1 6 s 6 k þ 1 and W s Hs1 ðXÞ. Let ðuh ; ph Þ be the finite element approximation of the solution ðu; pÞ, then
kp Rs ph k 6 Ch where
aðtþ1Þ
.
kpktþ1 þ Ch ðjjju uh jjj þ kp ph kÞ;
ð3:19Þ
. ¼ s 1 þ a minð0; 1 sÞ.
Proof. Since Rs is the L2 -projection, then
kp Rs pk 6 C stþ1 kpktþ1 ¼ Ch
aðtþ1Þ
kpktþ1 :
ð3:20Þ
Combining (3.20) and (3.8) gives
kp Rs ph k 6 kp Rs pk þ kRs p Rs ph k 6 Ch
aðtþ1Þ
kpktþ1 þ Ch
s1þa minð0;1sÞ
ðjjju uh jjj þ kp ph kÞ:
ð3:21Þ
This completes the proof of the lemma. h Using (2.10) we can bound the right-hand side of (3.12) and (3.19) as follow: a
ku Q s uh k þ h krs ðu Q s uh Þk 6 Ch and
kp Rs ph k 6 Ch
aðtþ1Þ
kpktþ1 þ Ch
.þk
aðrþ1Þ
kukrþ1 þ Ch
rþk
ðkukkþ1 þ kpkk Þ
kukkþ1 þ kpkk ;
ð3:22Þ
ð3:23Þ
where r and . are defined as in Theorems 3.1 and 3.2. The velocity estimate can be optimized by choosing a ¼ au such that
au ðr þ 1Þ ¼ k þ s 1 þ au minð0; 2 sÞ:
ð3:24Þ
The corresponding error estimate is given by a
ku Q s uh k þ h u krs ðu Q s uh Þk 6 Ch
au ðrþ1Þ
kukrþ1 þ kukkþ1 þ kpkk :
ð3:25Þ
Similarly, the pressure estimate can be optimized by choosing a ¼ ap such that
ap ðt þ 1Þ ¼ k þ s 1 þ ap minð0; 1 sÞ:
ð3:26Þ
The corresponding error estimate for the post-processed pressure approximation is given by
kp Rs ph k 6 Ch
ap ðtþ1Þ
kpktþ1 þ kukkþ1 þ kpkk :
ð3:27Þ
The results are summarized as follows. Theorem 3.3. Suppose that (2.13) holds true with 1 6 s 6 k þ 1. Let the surface fitting spaces V s and W s be sufficiently smooth such that V s ðHs2 ðXÞÞ2 and W s Hs1 ðXÞ. Let ðuh ; ph Þ be the finite element approximation of the solution ðu; pÞ. Then, the post-processed velocity approximation is estimated by (3.25) with
au ¼
kþs1 : r þ 1 minð0; 2 sÞ
ð3:28Þ
In addition, the postprocessed pressure is estimated by (3.27) with
ap ¼
kþs1 : t þ 1 minð0; 1 sÞ
ð3:29Þ
4. Numerical experiments In this section, we present several numerical experiments for supporting our theoretical analysis. All the numerical examples are defined on domain ½0; 1 ½0; 1 with uniform triangulations. The triangulation T h is constructed by: (1) dividing the domain into an n n rectangular mesh; (2) connecting the diagonal line with the negative slope. Denote h ¼ 1=n as the mesh size. The Hðdiv Þ finite element spaces are defined by
V h ¼ fv 2 Hðdiv Þ : v jK 2 BDM 1 ðKÞ; W h ¼ fq 2
L20 ð
XÞ : qjK 2 P0 ðKÞ;
8K 2 T h g;
8K 2 T h g:
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In the implementation, the quantities Q s uh and Rs ph are L2 -projection of uh and ph to V s and W s , respectively associated with T s . We define V s and W s as follows:
V s ¼ fv 2 ðL2 ðXÞÞ2 : v jK 2 ðP2 ðKÞÞ2 ; 1
W s ¼ fq 2 H ðXÞ : qjK 2 P 1 ðKÞ;
8K 2 T s g;
8K 2 T s g:
The following numerical results agree with the theory. Example 1. Let the domain X ¼ ½0; 1 ½0; 1. The exact solution for the velocity is assumed as
" u¼
2x2 yðx 1Þ2 ð2y 1Þðy 1Þ
#
2xy2 ð2x 1Þðx 1Þðy 1Þ2
and the pressure is assumed as
pðy xÞ p ¼ sin : 2 Table 1 shows that after the postprocessing methods, all the errors are reduced. The velocity in L2 -norm of ku Q s uh k has 2 the similar convergence rate as ku uh k, which is shown as Oðh Þ. There is no improvement for the velocity in L2 -norm. However, the error of velocity in H1 -norm and the error of pressure in L2 -norm have higher convergence rate, which are 1:5 0:5 shown as Oðh Þ. The order of convergence rate is Oðh Þ better than krh ðu uh Þk and kp ph k, respectively. All of these results establish highly agreement with the theoretical conclusion. Example 2. Let the domain X ¼ ½0; 1 ½0; 1, the exact solution for the velocity is assumed as
" u¼
10x2 yðx 1Þ2 ð2y 1Þðy 1Þ
#
10xy2 ð2x 1Þðx 1Þðy 1Þ2
and the pressure is assumed as
p ¼ 10ð2x 1Þð2y 1Þ: From the results shown in Table 2, it is clear that both the velocity in H1 -norm and pressure in L2 -norm have the superconvergence, but there is no improvement for the velocity in the L2 -norm. This agrees well with our theory. Example 3. Let the domain X ¼ ½0; 1 ½0; 1, the exact solution for the velocity is assumed as
u¼
cosð2pxÞsinð2pyÞ
sinð2pxÞcosð2pyÞ
Table 1 Errors on uniform triangular meshes T h and T s . h
ju uh j1
ku uh k
kp ph k
ju Q s uh j1
ku Q s uh k
kp Rs ph k
23
0.4914e2
0.7964e4
0.2002e1
0.1009e2
0.7629e4
0.2073e2
33
0.4333e2
0.6235e4
0.1768e1
0.8247e3
0.5973e4
0.1696e2
43
0.3876e2
0.5012e4
0.1584e1
0.6931e3
0.4802e4
0.1419e2
53
0.3505e2
0.4115e4
0.1434e1
0.5958e3
0.3943e4
0.1210e3
63 k
Oðh Þ; k ¼
0.3200e2
0.3439e4
0.1310e1
0.5215e3
0.3296e4
0.1048e3
1.0021
1.9795
0.9955
1.3535
1.9787
1.5742
Table 2 Errors on uniform triangular meshes T h and T s . h
ju uh j1
ku uh k
kp ph k
ju Q s uh j1
ku Q s uh k
kp Rs ph k
23
0.2457e1
0.3982e3
0.1358
0.5048e2
0.3814e3
0.1897e1
33
0.2046e1
0.2787e3
0.1133
0.3768e2
0.2670e3
0.1394e1
43
0.1752e1
0.2057e3
0.9718e1
0.2979e2
0.1971e3
0.1077e1
53
0.1533e1
0.1581e3
0.8506e1
0.2453e2
0.1515e3
0.8630e2
63
0.1362e1
0.1252e3
0.7563e1
0.2083e2
0.1200e3
0.7109e2
1.0022
1.9789
0.9974
1.3619
1.9781
1.5401
k
Oðh Þ; k ¼
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Table 3 Errors on uniform triangular meshes T h and T s . h
ju uh j1
ku uh k
kp ph k
ju Q s uh j1
ku Q s uh k
kp Rs ph k
3
2
0.2448
0.2667e2
0.6271
0.4028e1
0.2612e2
0.4371e1
33
0.2254
0.2265e2
0.5781
0.3588e1
0.2218e2
0.3829e1
43
0.2088
0.1948e2
0.5361
0.3234e1
0.1907e2
0.3389e1
53
0.1945
0.1693e2
0.4999
0.2943e1
0.1657e2
0.3027e1
63
0.1821
0.1484e2
0.4682
0.2700e1
0.1454e2
0.2724e1
1.0033
1.9877
0.9916
1.3434
1.9876
1.5022
k
Oðh Þ; k ¼
and the pressure is assumed as
p ¼ x þ y 1: 1:5
Table 3 gives the errors profile for Example 3. Notice that, the gradient estimate is of order Oðh Þ, that is much better than the optimal order OðhÞ. Likewise the pressure in the L2 -norm has the similar improvement, although there is no improvement for the velocity in the L2 -norm. Also, the numerical results and theoretical conclusions show highly consistent. Thus, it validate the superconvergence theories. References [1] I. Babus˘ka, The finite element method with Lagrangian multiplier, Numer. Math. 20 (1973) 179–192. [2] F. Brezzi, On the existence, uniqueness, and approximation of saddle point problems arising from Lagrangian multipliers, R.A.I.R.O., Anal. Numér. 2 (1974) 129–151. [3] P.G. Ciarlet, The Finite Method for Elliptic Problems, North-Holland, New York, 1978. [4] B. Cockburn, G. Kanschat, D. Schötzau, C. Schwab, Local discontinuous Galerkin methods for the Stokes system, SIAM J. Numer. Anal. 40 (2002) 319–343. [6] J. Douglas Jr., J. Wang, A superconvergence for mixed finite element methods on rectangular domains, Calcolo 26 (1989) 121–134. [7] R.E. Ewing, M. Liu, J. Wang, Superconvergence of mixed finite element approximations over quadrilaterals, SIAM J. Numer. Anal. 36 (3) (1998) 772–787. [8] V. Girault, P.A. Raviart, Finite Element Methods for the Navier–Stokes Equations: Theory and Algorithms, Springer, Berlin, 1986. [9] M.D. Gunzburger, Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice and Algorithms, Academic Press, San Diego, 1989. [11] R. Lazarov, A.B. Andreev, M. Hatri, Superconvergence of the gradients in the finite element method for some elliptic and parabolic problems, Variational-difference methods in mathematical physics, Part II, in: Proceedings of the fifth International Conference, Moscow, 1984, pp. 13–25. [12] B. Li, Z. Zhang, Analysis of a class of superconvergence patch recovery techniques for linear and bilinear finite elements, Numer. Methods Partial Differ. Equ. 15 (1999) 151–167. [13] Q. Lin, J. Li, A. Zhou, A rectangle test for the Stokes problem, in: Proceedings of System Science and Systems Engineering, Great Hall (H.K.) Culture Publishing Co., 1991, pp. 236–237. [16] J. Pan, Global superconvergence for the bilinear-constant scheme for the Stokes problem, SIAM J. Numer. Anal. 34 (1997) 2424–2430. [17] A.H. Schatz, I.H. Sloan, L.B. Wahlbin, Superconvergence in finite element methods and meshes that are symmetric with respect to a point, SIAM J. Numer Anal. 33 (1996) 505–521. [18] X. Tai, J. Wang, Superconvergence for the gradient of finite element approximations by least-squares surface fitting, SIAM J. Numer. Anal. 40 (2002) 1263–1280. [20] J. Wang, A superconvergence analysis for finite element solutions by the least-squares surface fitting on irregular meshes for smooth problems, J. Math. Study 33 (3) (2000) 229–243. [21] J. Wang, X. Ye, Superconvergence of finite element approximations for the Stokes problem by projection methods, SIAM J. Numer. Anal. 39 (2001) 1001–1013. [22] J. Wang, X. Ye, The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique, Int. J. Numer. Methods. Eng. 33 (1992) 1331–1364. [23] Z. Zhang, J.Z. Zhu, Analysis of the superconvergent patch recovery technique and a posteriori error estimator in the finite element method (II), Comput. Methods Appl. Mech. Eng. 163 (1998) 159–170. [24] O.C. Zienkiewicz, J.Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. Parts 1: The recovery technique, Int. J. Numer. Methods. Eng. 33 (1992) 1331–1364. [25] O.C. Zienkiewicz, J.Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. Parts 2: Error estimates and adaptivity, Int. J. Numer. Methods Eng. 33 (1992) 1365–1382. [26] M. Zlamal, Superconvergence and reduced integration in the finite element method, Math. Comput. 32 (1977) 663–685. [27] J. Wang, X. Ye, New finite element methods in computational fluid dynamics by H(div) elements, SIAM J. Numer. Anal. 45 (2007) 1269–1286. [28] P. Raviart, J. Thomas, A mixed finite element method for second order elliptic problems, in: I. Galligani, E. Magenes (Eds.), Mathematical Aspects of the Finite Element Method, Lectures Notes in Math., vol. 606, New York, 1977. [29] J. Wang, Y. Wang, X. Ye, A robust numerical methods for Stokes equations based on divergence-free Hðdiv Þ finite element methods, SIAM J. Sci. Comput. 31 (4) (2009) 2784–2802. [30] J. Claes, Numerical Solution of partial Differential Equations by the Finite Element Method, Dover, 2009.
Further Reading [5] M. Crouzeix, P.A. Raviart, Conforming and non-conforming finite element methods for solving the stationary Stokes equations, R.A.I.R.O. R3 (1973) 33– 76. [10] W. Hoffmann, A.H. Schatz, L.B. Wahlbin, G. Wittum, Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. Part 1: A smooth problem and globally quasi-uniform meshes, Math. Comput. 70 (2001) 897–909. [14] Q. Lin, J. Pan, Global superconvergence for rectangular elements for the Stokes problem, in: Proceedings of System Science and Systems Engineering, Great Hall (H.K.) Culture Publishing Co., 1991, pp. 371–376. [15] Q. Lin, N. Yan, Analysis and Construction of Finite Element Methods with High Efficiency, Hebei University Publishing, 1996 (in Chinese). [19] L.B. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Mathematics, vol. 1605, Springer, Berlin, 1995.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Superconvergence of Hðdiv Þ finite element approximations for the Stokes problem by L2 -projection methods Rabeea Jari a,⇑, Lin Mu a, Xiu Ye b a b
Department of Applied Science, University of Arkansas at Little Rock, 2801 South University, Little Rock, AR 72204, United States Department of Mathematics, University of Arkansas at Little Rock, 2801 South University, Little Rock, AR 72204, United States
a r t i c l e
i n f o
Keywords: Finite element methods Superconvergence L2 -projection The Stokes equations
a b s t r a c t A general superconvergence result of Hðdiv Þ finite element approximations for the Stokes equations is established by using L2 -projection method. Regularity assumptions for the Stokes problem with regular partitions is required. Numerical experiments are given to verify the theoretical results. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction The superconvergence of finite element solutions is an interesting and useful phenomenon in scientific computing of real world problems. In recent years, the superconvergence for finite element solutions has been an active research area in numerical analysis [6,7,11,12,16–18,22–26]. The purpose of this paper is to improve the existing approximation accuracy by applying certain postprocessing techniques which is easy to implement with little cost. L2 -projection method was proposed and analyzed to gain superconvergence by Wang [20] for the second order elliptic problem. It has been extended by Wang and Ye [21] to the Stokes equations for H1 conforming finite element method [8,9,13]. This paper aims at a study of superconvergence by using the L2 -projection method to the Hðdiv Þ finite element for the Stokes equations. The superconvergence result is based on some regularity assumption of the exact solution for the Stokes problem. The continuity equation r uh ðxÞ ¼ 0 8x 2 X requires the numerical solution uh to be a member of the Sobolev space Hðdiv Þ. To satisfy this property, several finite element schemes have been developed to generate locally divergence-free solutions [4]. Particularly, a recent approach by using Hðdiv Þ conforming finite elements method has been proposed and studied for incompressible fluid flow problems [27]. The main advantage of Hðdiv Þ finite element methods is that the discrete velocity field is exactly divergence-free. Hence, it seems that the Hðdiv Þ finite element might be easily decoupled between the velocity and the pressure unknowns, since this methods satisfy the continuity equation and the inf–sup condition of Brezzi [2] and Babuška [1]. The remainder of the paper is organized as follows: in Section 2, the Stokes equations is presented, along with variational formulation of Hðdiv Þ finite element for the Stokes equations and its finite element scheme. In Section 3, we establish superconvergence error estimates for both velocity and pressure approximations of Hðdiv Þ finite element for the Stokes problem by using L2 -projection method. Finally, in Section 4, several numerical experiments are present to support the theoretical conclusion.
⇑ Corresponding author. E-mail addresses: [email protected] (R. Jari), [email protected] (L. Mu), [email protected] (X. Ye). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.11.023
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2. Preliminaries for the Stokes problem Let us consider the Stokes equations with the homogeneous boundary condition for the velocity variable. Let X be a bounded, connected, and open subset of R2 . The Stokes problem seeks the velocity u and the pressure p in a certain functional spaces such that
mDu þ rp ¼ f in X; r u ¼ 0 in X; u ¼ 0 on @ X;
ð2:1Þ ð2:2Þ ð2:3Þ
where m denotes the fluid viscosity; D; r, and r denote the Laplacian, gradient, and divergence operators, respectively; f ¼ fðxÞ 2 ðL2 ðXÞÞ2 is the unit external volumetric force acting on the fluid at x 2 X [28,30]. In this paper, we assume that m ¼ 1. This paper will use the standard notations for the Sobolev spaces Hs ðXÞ and their associated inner products ð; Þs , norms k ks , and semi-norms j js for s P 0. The Sobolev space H0 ðXÞ coincides with the space of square integrable functions L2 ðXÞ, in which case the norm and inner product are denoted by k k and ð; Þ, respectively. Moreover, denote by L20 ðXÞ the subspace of L2 ðXÞ consisting of the functions in L2 ðXÞ vanishing mean value. The space Hðdiv Þ is a set of vector-valued functions on X defined as follows,
Hðdiv Þ ¼ fv : v 2 ðL2 ðXÞÞ2 ; r v 2 L2 ðXÞg equipped with the norm 1
kv kHðdiv Þ ¼ ðkv k2 þ kr v k2 Þ2 : Let T h be a finite element partition of the domain X with characteristic mesh size h, assume that the partition T h is quasiuniform: i.e, it is shape regular and satisfies the inverse assumption [3]. Define finite element spaces V h and W h for the velocity and pressure, respectively, by
V h ¼ fv 2 Hðdiv Þ : v jK 2 V k ðKÞ;
8K 2 T h ;
W h ¼ fq 2 L20 ðXÞ : qjK 2 W m ðKÞ;
8K 2 T h g;
v nj@X ¼ 0g;
where n is the outward normal direction on the boundary of X; V k ðKÞ is a space of vector-valued polynomials on the element K with index k P 1, and W m ðKÞ is a space of polynomials on the element K with index m P 0. Let e be an interior edge shared by two elements K 1 and K 2 , and let n1 and n2 be unit normal vectors on e pointing exterior to K 1 and K 2 , respectively. Denote by s1 and s2 the two unit tangential vectors which make the right-hand coordinate systems with n1 and n2 , respectively. We define the average fg and jump ½ on e for vector-valued functions w as follows:
feðwÞg ¼
1 ðn1 rðw s1 Þj@K 1 þ n2 rðw s2 Þj@K 2 Þ 2
and
½w ¼ wj@K 1 s1 þ wj@K 2 s2 : If e is an edge on the boundary X with e 2 @K 1 \ @K 2 , the above two operations modified by
feðwÞg ¼ n1 rðw s1 Þj@K 1 ; ½w ¼ wj@K 1 s1 :
v
Let E h denote the union of the boundaries of all elements K 2 T h . Let VðhÞ ¼ V h þ ðHs ðXÞ \ H10 ðXÞÞ2 , with s > 32. For 2 VðhÞ, define rh v to be the function whose restriction to each element K 2 T h is given by the standard gradient rv . The Hðdiv Þ finite element approximation problem of (2.1)–(2.3) seeks ðuh ; ph Þ 2 V h W h such that
aðuh ; v Þ bðv ; ph Þ ¼ ðf; v Þ; bðuh ; qÞ ¼ 0;
8v 2 V h ;
ð2:4Þ
8q 2 W h ;
ð2:5Þ
with
aðuh ; v Þ ¼ ðrh uh ; rh v Þ
XZ e2eh
and
bðuh ; qÞ ¼ ðr uh ; qÞ;
e
feðuh Þg½v ds þ
XZ e2eh
e
1
ðdhe ½uh ½v feðv Þg½uh Þds
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R. Jari et al. / Applied Mathematics and Computation 219 (2013) 5649–5656
where d is a positive parameter that should be large enough to guarantee a good convergence rate and as small as possible in order to keep the condition number of the discrete system low, (see [29]), and he is the length of the edge e. The exact solution ðu; pÞ of the Stokes problem belongs to VðhÞ satisfies the following variational equations:
aðu; v Þ bðv ; pÞ ¼ ðf; v Þ; bðu; qÞ ¼ 0;
8 v 2 V h;
ð2:6Þ
8 q 2 W h:
ð2:7Þ
We introduce two norms jjj jjj and jjj jjj1 for the set VðhÞ as follows:
jjjv jjj21 ¼ jv j21;h þ
X 1 he k½v k2e ;
ð2:8Þ
e2eh
jjjv jjj2 ¼ jjjv jjj21 þ
X
he kfeðv Þgk2e ;
ð2:9Þ
e2eh
where
jv j21;h ¼
X
jv j21;K and kv k2e ¼
K2T h
Z
v v ds:
e
The following lemma and theorem can be found in [27]. Lemma 2.1. There exists a constant C independent of h such that
jaðw; v Þj 6 Cjjjwjjj jjjv jjj; jbðw; qÞj 6 Cjjjwjjj kqk;
8w;
v 2 VðhÞ;
8w 2 VðhÞ;
q 2 L20 ðXÞ:
Theorem 2.1. Let ðu; pÞ be the solution of (2.1)–(2.3) and ðuh ; ph Þ 2 V h W h be the solution of (2.4) and (2.5), there exists a constant C independent of h such that k
jjju uh jjj þ kp ph k 6 Ch ðkukkþ1 þ kpkk Þ:
ð2:10Þ
In the next section, the superconvergence analysis requires certain regularity for the Stokes problem. To this end, we consider a more general Stokes problem: find ðu; pÞ 2 ðH10 ðXÞÞ2 L20 ðXÞ satisfying
8v 2 ðH10 ðXÞÞ2 ;
aðu; v Þ bðv ; pÞ ¼ ðf; v Þ; bðu; qÞ ¼ ðg; qÞ;
8q 2
ð2:11Þ
L20 ð
XÞ;
L20 ð
ð2:12Þ s
where g 2 XÞ is a given function. Assume that the domain X is so regular that ensures a H ; s P 1 regularity for the solution of (2.11) and (2.12). In other words, for any f 2 ðHs2 ðXÞÞ2 and g 2 Hs1 ðXÞ \ L20 ðXÞ, the problem (2.11) and (2.12) has a unique solution u 2 ðH10 ðXÞÞ2 \ ðHs ðXÞÞ2 and p 2 Hs1 ðXÞ \ L20 ðXÞ satisfying the following a priori estimate:
kuks þ kpks1 6 Cðkfks2 þ kgks1 Þ;
ð2:13Þ
where C is a constant independent of the data f and g, [21]. 3. Superconvergence by L2 -projection The L2 -projection technique was introduced by Wang [20]. It projects the approximate solution to another finite element dimensional space associated with a coarse mesh. The difference of the size in the two mesh sizes can be used to achieve a superconvergence. In order to extend the idea in [20] to the Hðdiv Þ finite element approximation for the Stokes problem, we start with defining a coarse mesh T s where s h satisfying:
s ¼ ha ;
ð3:1Þ
with a 2 ð0; 1Þ. For now, the parameter a will play an important role in the postprocessing. Define two finite element spaces V s ðHs2 ðXÞÞ2 and W s Hs1 ðXÞ for the velocity and pressure, respectively. Here V s and W s are consisting of piecewise polynomials of degree r and t, respectively associated with the partition T s . Let Q s and Rs to be the L2 -projectors from L2 ðXÞ onto the finite element spaces V s and W s , respectively. In the following, we will analyze the error of u Q s uh and p Rs ph . The following lemma provides an error estimate for Q s u Q s uh . Lemma 3.1. Suppose that (2.13) holds with 1 6 s 6 k þ 1 and V s ðHs2 ðXÞÞ2 . Then there is a constant C independent of h and s such that
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kQ s u Q s uh k 6 Ch
s1þa minð0;2sÞ
ðjjju uh jjj þ kp ph kÞ;
ð3:2Þ
where a 2 ð0; 1Þ is as defined in (3.1). Proof. Using the definition of k k and Q s , we have
kQ s u Q s uh k ¼
sup
jðQ s u Q s uh ; /Þj:
/2ðL2 ðXÞÞ2 ;k/k¼1
From the definition of the L2 -projection for Q s , we have
ðQ s u Q s uh ; /Þ ¼ ðu uh ; Q s /Þ: It follows directly that
kQ s u Q s uh k ¼
sup
jðu uh ; Q s /Þj:
ð3:3Þ
/2ðL2 ðXÞÞ2 ;k/k¼1
Consider the following problem: find ðw; kÞ 2 ðH10 ðXÞÞ2 L20 ðXÞ such that
8v 2 ðH10 ðXÞÞ2 ;
aðw; v Þ bðv ; kÞ ¼ ðQ s /; v Þ; bðw; qÞ ¼ 0;
ð3:4Þ
8 q 2 L20 ðXÞ:
ð3:5Þ
The difference of (2.4) and (2.6) gives
aðu uh ; v Þ bðv ; p ph Þ ¼ 0;
8v 2 V h :
ð3:6Þ
The difference of (2.5) and (2.7) yields
bðu uh ; qÞ ¼ 0; By setting
8 q 2 W h:
v in (3.4) by u uh and then using (3.6) and Lemma 2.1, we obtain
ðu uh ; Q s /Þ ¼ aðw; u uh Þ bðu uh ; kÞ ¼ aðw wI ; u uh Þ þ bðwI ; p ph Þ bðu uh ; k kI Þ ¼ aðw wI ; u uh Þ bðw wI ; p ph Þ bðu uh ; k kI Þ 6 Cðjjjw wI jjj jjju uh jjj þ jjjw wI jjj kp ph k þ jjju uh jjj kk kI kÞ;
ð3:7Þ
where wI 2 V h and kI 2 W h are two interpolation functions. Using the Schwarz inequality and (2.10) we have
jðu uh ; Q s /Þj 6 Cðjjjw wI jjj þ kk kI kÞðjjju uh jjj þ kp ph kÞ 6 Ch
s1
ðkwks þ kkks1 Þðjjju uh jjj þ kp ph kÞ:
In addition, from (2.13), the inverse inequality and (3.1) we obtain
jðu uh ; Q s /Þj 6 Ch 6 Ch
s1
kQ s /ks2 ðjjju uh jjj þ kp ph kÞ 6 Ch
s1þa minð0;2sÞ
s1 minð0;2sÞ
s
ðjjju uh jjj þ kp ph kÞk/k
ðjjju uh jjj þ kp ph kÞk/k:
The last estimate, together with (3.3), completes the proof of the lemma. h The following lemma provides an error estimate for Rs p Rs ph . Lemma 3.2. Suppose that (2.13) holds with 1 6 s 6 k þ 1 and W s Hs1 ðXÞ. Then there is a constant C independent of h and s such that
kRs p Rs ph k 6 Ch
s1þa minð0;1sÞ
ðjjju uh jjj þ kp ph kÞ;
ð3:8Þ
where a 2 ð0; 1Þ is as defined in (3.1). Proof. From the definition of k k and Rs , we have
kRs p Rs ph k ¼
sup
jðRs p Rs ph ; /Þj:
/2L20 ðXÞ;k/k¼1
From the definition of the L2 -projection for Rs , we have
ðRs p Rs ph ; /Þ ¼ ðp ph ; Rs /Þ: It follows directly that,
kRs p Rs ph k ¼
sup
jðp ph ; Rs /Þj:
/2L20 ðXÞ;k/k¼1
ð3:9Þ
R. Jari et al. / Applied Mathematics and Computation 219 (2013) 5649–5656
5653
Consider the following problem: find ðx; nÞ 2 ðH10 ðXÞÞ2 L20 ðXÞ such that
8v 2 ðH10 ðXÞÞ2 ;
aðx; v Þ bðv ; nÞ ¼ 0; bðx; qÞ ¼ ðRs /; qÞ;
8q 2
L20 ð
ð3:10Þ
XÞ:
ð3:11Þ
Setting q in (3.11) by p ph and using (3.6), we obtain
ðp ph ; Rs /Þ ¼ bðx; p ph Þ ¼ bðx xI ; p ph Þ þ bðxI ; p ph Þ ¼ bðx xI ; p ph Þ þ aðu uh ; xI Þ ¼ bðx xI ; p ph Þ þ aðu uh ; xI xÞ þ aðu uh ; xÞ ¼ bðx xI ; p ph Þ þ aðu uh ; xI xÞ þ bðu uh ; nÞ ¼ bðx xI ; p ph Þ þ aðu uh ; xI xÞ þ bðu uh ; n nI Þ; where xI 2 V h and nI 2 W h are two interpolation functions. Using the Schwarz inequality we obtain
jðp ph ; Rs /Þj 6 C ðjjjx xI jjjkp ph k þ jjju uh jjjjjjx xI jjj þ kn nI kjjju uh jjjÞ 6 Cðjjjx xI jjj þ kn nI kÞðjjju uh jjj þ kp ph kÞ: Using (2.10) then,
jðp ph ; Rs /Þj 6 Ch
s1
ðkxks þ knks1 Þðjjju uh jjj þ kp ph kÞ:
Furthermore, from (2.13) we have,
jðp ph ; Rs /Þj 6 Ch 6 Ch
s1
kRs /ks1 ðjjju uh jjj þ kp ph kÞ 6 Ch
s1þa minð0;1sÞ
s1 minð0;1sÞ
s
ðjjju uh jjj þ kp ph kÞk/k
ðjjju uh jjj þ kp ph kÞk/k:
The last estimate, together with (3.9), completes the proof of the lemma. Now, we can estimate u Q s uh .
h
Theorem 3.1. Assume that (2.13) holds true with 1 6 s 6 k þ 1 and V s ðHs2 ðXÞÞ2 . If ðuh ; ph Þ is the finite element approximation of the solution ðu; pÞ, then we have a
ku Q s uh k þ h krs ðu Q s uh Þk 6 Ch where
aðrþ1Þ
r
kukrþ1 þ Ch ðjjju uh jjj þ kp ph kÞ;
ð3:12Þ
r ¼ s 1 þ a minð0; 2 sÞ.
Proof. Since Q s is the L2 -projection, then
ku Q s uk 6 C srþ1 kukrþ1 6 Ch
aðrþ1Þ
kukrþ1 :
ð3:13Þ
Combining (3.13) and (3.2) gives
ku Q s uh k 6 ku Q s uk þ kQ s u Q s uh k 6 Ch
aðrþ1Þ
kukrþ1 þ Ch
s1þa minð0;2sÞ
ðjjju uh jjj þ kp ph kÞ;
which completes the estimate for ku Q s uh k in (3.12). Next, we estimate krs ðu Q s uh Þk as follows. Since Q s is the L2 -projection, then ar
krs ðu Q s uÞk 6 C sr kukrþ1 6 Ch kukrþ1 :
ð3:14Þ
a
Multiplying both sides of (3.14) by h , we get a
aðrþ1Þ
h krs ðu Q s uÞk 6 Ch
kukrþ1 :
ð3:15Þ
Moreover, the inverse inequality gives
krs ðQ s u Q s uh Þk 6 C s1 kQ s u Q s uh k:
ð3:16Þ
Using (3.1), the above inequality implies a
h krs ðQ s u Q s uh Þk 6 CkQ s u Q s uh k:
ð3:17Þ
Consequently, from (3.2), we obtain a
h krs ðQ s u Q s uh Þk 6 Ch
s1þa minð0;2sÞ
ðjjju uh jjj þ kp ph kÞ:
ð3:18Þ
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R. Jari et al. / Applied Mathematics and Computation 219 (2013) 5649–5656
Combining (3.15) and (3.18) gives a
a
a
h krs ðu Q s uh Þk 6 h krs ðu Q s uÞk þ h krs ðQ s u Q s uh Þk 6 Ch
aðrþ1Þ
kukrþ1 þ Ch
s1þa minð0;2sÞ
ðjjju uh jjj þ kp ph kÞ:
This completes the proof of the theorem. h Likewise, we can derive a similar result for the pressure. Theorem 3.2. Suppose that (2.13) holds true with 1 6 s 6 k þ 1 and W s Hs1 ðXÞ. Let ðuh ; ph Þ be the finite element approximation of the solution ðu; pÞ, then
kp Rs ph k 6 Ch where
aðtþ1Þ
.
kpktþ1 þ Ch ðjjju uh jjj þ kp ph kÞ;
ð3:19Þ
. ¼ s 1 þ a minð0; 1 sÞ.
Proof. Since Rs is the L2 -projection, then
kp Rs pk 6 C stþ1 kpktþ1 ¼ Ch
aðtþ1Þ
kpktþ1 :
ð3:20Þ
Combining (3.20) and (3.8) gives
kp Rs ph k 6 kp Rs pk þ kRs p Rs ph k 6 Ch
aðtþ1Þ
kpktþ1 þ Ch
s1þa minð0;1sÞ
ðjjju uh jjj þ kp ph kÞ:
ð3:21Þ
This completes the proof of the lemma. h Using (2.10) we can bound the right-hand side of (3.12) and (3.19) as follow: a
ku Q s uh k þ h krs ðu Q s uh Þk 6 Ch and
kp Rs ph k 6 Ch
aðtþ1Þ
kpktþ1 þ Ch
.þk
aðrþ1Þ
kukrþ1 þ Ch
rþk
ðkukkþ1 þ kpkk Þ
kukkþ1 þ kpkk ;
ð3:22Þ
ð3:23Þ
where r and . are defined as in Theorems 3.1 and 3.2. The velocity estimate can be optimized by choosing a ¼ au such that
au ðr þ 1Þ ¼ k þ s 1 þ au minð0; 2 sÞ:
ð3:24Þ
The corresponding error estimate is given by a
ku Q s uh k þ h u krs ðu Q s uh Þk 6 Ch
au ðrþ1Þ
kukrþ1 þ kukkþ1 þ kpkk :
ð3:25Þ
Similarly, the pressure estimate can be optimized by choosing a ¼ ap such that
ap ðt þ 1Þ ¼ k þ s 1 þ ap minð0; 1 sÞ:
ð3:26Þ
The corresponding error estimate for the post-processed pressure approximation is given by
kp Rs ph k 6 Ch
ap ðtþ1Þ
kpktþ1 þ kukkþ1 þ kpkk :
ð3:27Þ
The results are summarized as follows. Theorem 3.3. Suppose that (2.13) holds true with 1 6 s 6 k þ 1. Let the surface fitting spaces V s and W s be sufficiently smooth such that V s ðHs2 ðXÞÞ2 and W s Hs1 ðXÞ. Let ðuh ; ph Þ be the finite element approximation of the solution ðu; pÞ. Then, the post-processed velocity approximation is estimated by (3.25) with
au ¼
kþs1 : r þ 1 minð0; 2 sÞ
ð3:28Þ
In addition, the postprocessed pressure is estimated by (3.27) with
ap ¼
kþs1 : t þ 1 minð0; 1 sÞ
ð3:29Þ
4. Numerical experiments In this section, we present several numerical experiments for supporting our theoretical analysis. All the numerical examples are defined on domain ½0; 1 ½0; 1 with uniform triangulations. The triangulation T h is constructed by: (1) dividing the domain into an n n rectangular mesh; (2) connecting the diagonal line with the negative slope. Denote h ¼ 1=n as the mesh size. The Hðdiv Þ finite element spaces are defined by
V h ¼ fv 2 Hðdiv Þ : v jK 2 BDM 1 ðKÞ; W h ¼ fq 2
L20 ð
XÞ : qjK 2 P0 ðKÞ;
8K 2 T h g;
8K 2 T h g:
5655
R. Jari et al. / Applied Mathematics and Computation 219 (2013) 5649–5656
In the implementation, the quantities Q s uh and Rs ph are L2 -projection of uh and ph to V s and W s , respectively associated with T s . We define V s and W s as follows:
V s ¼ fv 2 ðL2 ðXÞÞ2 : v jK 2 ðP2 ðKÞÞ2 ; 1
W s ¼ fq 2 H ðXÞ : qjK 2 P 1 ðKÞ;
8K 2 T s g;
8K 2 T s g:
The following numerical results agree with the theory. Example 1. Let the domain X ¼ ½0; 1 ½0; 1. The exact solution for the velocity is assumed as
" u¼
2x2 yðx 1Þ2 ð2y 1Þðy 1Þ
#
2xy2 ð2x 1Þðx 1Þðy 1Þ2
and the pressure is assumed as
pðy xÞ p ¼ sin : 2 Table 1 shows that after the postprocessing methods, all the errors are reduced. The velocity in L2 -norm of ku Q s uh k has 2 the similar convergence rate as ku uh k, which is shown as Oðh Þ. There is no improvement for the velocity in L2 -norm. However, the error of velocity in H1 -norm and the error of pressure in L2 -norm have higher convergence rate, which are 1:5 0:5 shown as Oðh Þ. The order of convergence rate is Oðh Þ better than krh ðu uh Þk and kp ph k, respectively. All of these results establish highly agreement with the theoretical conclusion. Example 2. Let the domain X ¼ ½0; 1 ½0; 1, the exact solution for the velocity is assumed as
" u¼
10x2 yðx 1Þ2 ð2y 1Þðy 1Þ
#
10xy2 ð2x 1Þðx 1Þðy 1Þ2
and the pressure is assumed as
p ¼ 10ð2x 1Þð2y 1Þ: From the results shown in Table 2, it is clear that both the velocity in H1 -norm and pressure in L2 -norm have the superconvergence, but there is no improvement for the velocity in the L2 -norm. This agrees well with our theory. Example 3. Let the domain X ¼ ½0; 1 ½0; 1, the exact solution for the velocity is assumed as
u¼
cosð2pxÞsinð2pyÞ
sinð2pxÞcosð2pyÞ
Table 1 Errors on uniform triangular meshes T h and T s . h
ju uh j1
ku uh k
kp ph k
ju Q s uh j1
ku Q s uh k
kp Rs ph k
23
0.4914e2
0.7964e4
0.2002e1
0.1009e2
0.7629e4
0.2073e2
33
0.4333e2
0.6235e4
0.1768e1
0.8247e3
0.5973e4
0.1696e2
43
0.3876e2
0.5012e4
0.1584e1
0.6931e3
0.4802e4
0.1419e2
53
0.3505e2
0.4115e4
0.1434e1
0.5958e3
0.3943e4
0.1210e3
63 k
Oðh Þ; k ¼
0.3200e2
0.3439e4
0.1310e1
0.5215e3
0.3296e4
0.1048e3
1.0021
1.9795
0.9955
1.3535
1.9787
1.5742
Table 2 Errors on uniform triangular meshes T h and T s . h
ju uh j1
ku uh k
kp ph k
ju Q s uh j1
ku Q s uh k
kp Rs ph k
23
0.2457e1
0.3982e3
0.1358
0.5048e2
0.3814e3
0.1897e1
33
0.2046e1
0.2787e3
0.1133
0.3768e2
0.2670e3
0.1394e1
43
0.1752e1
0.2057e3
0.9718e1
0.2979e2
0.1971e3
0.1077e1
53
0.1533e1
0.1581e3
0.8506e1
0.2453e2
0.1515e3
0.8630e2
63
0.1362e1
0.1252e3
0.7563e1
0.2083e2
0.1200e3
0.7109e2
1.0022
1.9789
0.9974
1.3619
1.9781
1.5401
k
Oðh Þ; k ¼
5656
R. Jari et al. / Applied Mathematics and Computation 219 (2013) 5649–5656
Table 3 Errors on uniform triangular meshes T h and T s . h
ju uh j1
ku uh k
kp ph k
ju Q s uh j1
ku Q s uh k
kp Rs ph k
3
2
0.2448
0.2667e2
0.6271
0.4028e1
0.2612e2
0.4371e1
33
0.2254
0.2265e2
0.5781
0.3588e1
0.2218e2
0.3829e1
43
0.2088
0.1948e2
0.5361
0.3234e1
0.1907e2
0.3389e1
53
0.1945
0.1693e2
0.4999
0.2943e1
0.1657e2
0.3027e1
63
0.1821
0.1484e2
0.4682
0.2700e1
0.1454e2
0.2724e1
1.0033
1.9877
0.9916
1.3434
1.9876
1.5022
k
Oðh Þ; k ¼
and the pressure is assumed as
p ¼ x þ y 1: 1:5
Table 3 gives the errors profile for Example 3. Notice that, the gradient estimate is of order Oðh Þ, that is much better than the optimal order OðhÞ. Likewise the pressure in the L2 -norm has the similar improvement, although there is no improvement for the velocity in the L2 -norm. Also, the numerical results and theoretical conclusions show highly consistent. Thus, it validate the superconvergence theories. References [1] I. Babus˘ka, The finite element method with Lagrangian multiplier, Numer. Math. 20 (1973) 179–192. [2] F. Brezzi, On the existence, uniqueness, and approximation of saddle point problems arising from Lagrangian multipliers, R.A.I.R.O., Anal. Numér. 2 (1974) 129–151. [3] P.G. Ciarlet, The Finite Method for Elliptic Problems, North-Holland, New York, 1978. [4] B. Cockburn, G. Kanschat, D. Schötzau, C. Schwab, Local discontinuous Galerkin methods for the Stokes system, SIAM J. Numer. Anal. 40 (2002) 319–343. [6] J. Douglas Jr., J. Wang, A superconvergence for mixed finite element methods on rectangular domains, Calcolo 26 (1989) 121–134. [7] R.E. Ewing, M. Liu, J. Wang, Superconvergence of mixed finite element approximations over quadrilaterals, SIAM J. Numer. Anal. 36 (3) (1998) 772–787. [8] V. Girault, P.A. Raviart, Finite Element Methods for the Navier–Stokes Equations: Theory and Algorithms, Springer, Berlin, 1986. [9] M.D. Gunzburger, Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice and Algorithms, Academic Press, San Diego, 1989. [11] R. Lazarov, A.B. Andreev, M. Hatri, Superconvergence of the gradients in the finite element method for some elliptic and parabolic problems, Variational-difference methods in mathematical physics, Part II, in: Proceedings of the fifth International Conference, Moscow, 1984, pp. 13–25. [12] B. Li, Z. Zhang, Analysis of a class of superconvergence patch recovery techniques for linear and bilinear finite elements, Numer. Methods Partial Differ. Equ. 15 (1999) 151–167. [13] Q. Lin, J. Li, A. Zhou, A rectangle test for the Stokes problem, in: Proceedings of System Science and Systems Engineering, Great Hall (H.K.) Culture Publishing Co., 1991, pp. 236–237. [16] J. Pan, Global superconvergence for the bilinear-constant scheme for the Stokes problem, SIAM J. Numer. Anal. 34 (1997) 2424–2430. [17] A.H. Schatz, I.H. Sloan, L.B. Wahlbin, Superconvergence in finite element methods and meshes that are symmetric with respect to a point, SIAM J. Numer Anal. 33 (1996) 505–521. [18] X. Tai, J. Wang, Superconvergence for the gradient of finite element approximations by least-squares surface fitting, SIAM J. Numer. Anal. 40 (2002) 1263–1280. [20] J. Wang, A superconvergence analysis for finite element solutions by the least-squares surface fitting on irregular meshes for smooth problems, J. Math. Study 33 (3) (2000) 229–243. [21] J. Wang, X. Ye, Superconvergence of finite element approximations for the Stokes problem by projection methods, SIAM J. Numer. Anal. 39 (2001) 1001–1013. [22] J. Wang, X. Ye, The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique, Int. J. Numer. Methods. Eng. 33 (1992) 1331–1364. [23] Z. Zhang, J.Z. Zhu, Analysis of the superconvergent patch recovery technique and a posteriori error estimator in the finite element method (II), Comput. Methods Appl. Mech. Eng. 163 (1998) 159–170. [24] O.C. Zienkiewicz, J.Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. Parts 1: The recovery technique, Int. J. Numer. Methods. Eng. 33 (1992) 1331–1364. [25] O.C. Zienkiewicz, J.Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. Parts 2: Error estimates and adaptivity, Int. J. Numer. Methods Eng. 33 (1992) 1365–1382. [26] M. Zlamal, Superconvergence and reduced integration in the finite element method, Math. Comput. 32 (1977) 663–685. [27] J. Wang, X. Ye, New finite element methods in computational fluid dynamics by H(div) elements, SIAM J. Numer. Anal. 45 (2007) 1269–1286. [28] P. Raviart, J. Thomas, A mixed finite element method for second order elliptic problems, in: I. Galligani, E. Magenes (Eds.), Mathematical Aspects of the Finite Element Method, Lectures Notes in Math., vol. 606, New York, 1977. [29] J. Wang, Y. Wang, X. Ye, A robust numerical methods for Stokes equations based on divergence-free Hðdiv Þ finite element methods, SIAM J. Sci. Comput. 31 (4) (2009) 2784–2802. [30] J. Claes, Numerical Solution of partial Differential Equations by the Finite Element Method, Dover, 2009.
Further Reading [5] M. Crouzeix, P.A. Raviart, Conforming and non-conforming finite element methods for solving the stationary Stokes equations, R.A.I.R.O. R3 (1973) 33– 76. [10] W. Hoffmann, A.H. Schatz, L.B. Wahlbin, G. Wittum, Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. Part 1: A smooth problem and globally quasi-uniform meshes, Math. Comput. 70 (2001) 897–909. [14] Q. Lin, J. Pan, Global superconvergence for rectangular elements for the Stokes problem, in: Proceedings of System Science and Systems Engineering, Great Hall (H.K.) Culture Publishing Co., 1991, pp. 371–376. [15] Q. Lin, N. Yan, Analysis and Construction of Finite Element Methods with High Efficiency, Hebei University Publishing, 1996 (in Chinese). [19] L.B. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Mathematics, vol. 1605, Springer, Berlin, 1995.