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Finite element approximation for fourth-order nonlinear problem in the plane
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Applied Mathematics and Computation 194 (2007) 143–155 www.elsevier.com/locate/amc
Finite element approximation for fourth-order nonlinear problem in the plane Rong An *, Yuan Li, Kaitai Li Department of Mathematics, Xi’an Jiaotong University, Xi’an 710049, China
Abstract In this paper, a class of fourth-order nonlinear elliptic problem is investigated in a bounded convex domain X R2 . Under some assumptions, the existence and uniqueness of solution are proved via the Schaefer’s Fixed Point Theorem. Furthermore, conforming finite element approximation is applied and H2-error estimate and L2-error estimate are obtained. Finally, the numerical experiments are provided to verify our theoretical analysis. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Fourth-order nonlinear problem; Finite element approximation; Error estimates
1. Introduction The fourth-order elliptic problem is an important mathematical model in fluid mechanics [1], solid mechanics and other continuous physical system [2]. For example, in fluid mechanics, the stream function w of the incompressible flows in R2 satisfies the biharmonic problem; in solid mechanics, plate bending model is satisfied by harmonic problem, etc. Hence, large numbers of papers appear about this problem. As well known, the well-posedness theory of fourth-order linear problem is similar with second-order-problem [3]. Although many techniques like the maximum principle and comparison principle are not available for fourth-order problem with Dirichlet boundary, others still are valid, such as the methods of upper and lower solutions, and minimax method, in particular, if the equation is a Euler–Lagrange equation associated with some energy functional [4–6]. In addition, some fixed point theorems also are power tools for dealing with the nonlinear problem. We refer the reader to [7,8] for more details. In this paper, we will study a class of fourth-order nonlinear problem: Daðx; ru; DuÞ ¼ /ðjrujÞ þ f which comes from the mathematical model of the optimal shape design for blade’s surface of an impeller, which is a coupled system with nonlinear fourth-order elliptic problem and rotating Navier–Stokes equations *
Corresponding author. E-mail address: [email protected] (R. An).
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.04.060
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R. An et al. / Applied Mathematics and Computation 194 (2007) 143–155
[9]. Under some assumptions, the Schaefer’s Fixed Point Theorem is applied to prove the existence of solution. Comparing with other fixed point theorems, the advantage of Contracting Mapping Principle is that the solution not only exists but also is unique. Hence, in terms of Contracting Mapping Principle, we show that the solution is locally unique. In numerical analysis, the popular method for fourth-order problem is mixed finite element methods, such as [10–16] and references therein. Moreover, to our best knowledge, [16,17] are only two papers for fourth-order quasilinear problem. In [17], Karchevskii, Lyashko and Timerbaev discussed conforming finite element. But very strong assumptions were required. They assumed that the nonlinear operator of the problem was strongly monotone and Lipschitz continuous, from which the existence and uniqueness of weak solution and approximation solution were immediately obtained in an appropriate weighted Sobolev space. In this paper, we also investigate the finite element approximation using conforming element under some weak assumptions. It is well known that for linear biharmonic problem following error estimates hold (cf. [18]): ku uh kH 2 6 chðkukH 3 þ hkukH 4 Þ; ku uh kL2 6 ch3 ðkukH 3 þ hkukH 4 Þ; where c > 0 is independent of h. In this paper, the H2-error estimate is showed to be optimal via a linearized technique which is introduced and used to deal with the second-order nonlinear problem in [19]. However, the L2-error estimate is not optimal unless ay i ðx; y; zÞ 0, where y 2 R2 ; z 2 R; i ¼ 1; 2. This paper is organized as follows: in Section 2, we give some preliminary knowledge; in Section 3, the existence and uniqueness of solution are showed; in Section 4, the finite element subspace and the discretized variational formulation are given. Like Section 3, the approximation solution is locally unique; in Section 5, the H2-error estimate and L2-error estimate for sufficiently small h are obtained; in Section 6, the numerical experiments are provided to verify our theoretical analysis. 2. Preliminary Consider a class of fourth-order nonlinear problem: ( Daðx; ru; DuÞ ¼ /ðjrujÞ þ f in X; u ¼ ou ¼0 on
on oX;
ð1Þ
where uðxÞ : X ! R is unknown, aðx; y; zÞ : X R2 R ! R is a smooth function. Moreover, að; y; zÞ is nonlinear with respect to y and is linear with respect to z, /ðÞ : R ! R is a nonlinear operator, f is independent of u. X R2 is a bounded convex domain with boundary oX, n denotes the unite outer normal vector to oX. Introduce the linear space ov 2 2 ¼ 0 on oX V ¼ H 0 ðXÞ ¼ v 2 H ðXÞ; v ¼ on with the inner product Z Aðu; vÞ ¼ DuDv dx
8u; v 2 V
X
and the norm 1
kvkV ¼ Aðv; vÞ2
8v 2 V :
Following Sobolev inequalities are usually used: kukL1 6 c1 kukV ;
krukLq 6 c2 kukV
for all 2 6 q < þ1; 8u 2 V ;
where c1 and c2 depend only on X. Throughout this paper, we always assume the embedding constant c1 ¼ c2 ¼ 1.
R. An et al. / Applied Mathematics and Computation 194 (2007) 143–155
For f 2 V 0 ¼ H 2 ðXÞ, the variational formulation associated with the problem (1) is Find u 2 V such that R A1 ðu; vÞ ¼ X fv dx 8v 2 V ; where A1 ðu; vÞ ¼
Z
145
ð2Þ
aðx; ru; DuÞDv /ðjrujÞv dx:
X
Suppose that the nonlinear term aðx; y; zÞ and /ðsÞ satisfy m
p
(H1) aðx; y; zÞz P j0 jzj j1 jyj 1 , where m P 2, p1 > m, j0 and j1 both are positive constants and are independent of y; z and p1; p (H2) j/ðsÞj 6 j2 jsj 2 , where 2 6 p2 < 1; j2 is a positive constant and is independent of s and p2; p 1 p 2 2 (H3) / 2 C and j/0 ðsÞj 6 j3 jsj 2 , j/00 ðsÞj 6 j4 jsj 2 , where j3 and j4 both are positive constants and are independent of s and p2. Remark 1. From (H3), in view of mean-value theorem, we have j/ðs1 Þ /ðs2 Þj 6 j3 jas1 þ ð1 aÞs2 j
p2 1
p2 1
js1 s2 j 6 j3 ðjs1 j
p2 1
þ js2 j
Þjs1 s2 j;
ð3Þ
where 0 6 a 6 1. Remark 2. The condition p1 > m in (H1) can be replaced by p2 þ 1 > m in (H2). Following Schaefer’s Fixed Point Theorem plays a key role in the existence of solution. Lemma 1 [2, Theorem 9.4]. Let X be a real Banach space. Suppose that A : X ! X is a continuous and compact mapping. Assume further that the set fu 2 X ju ¼ kAðuÞ for some 0 6 k 6 1g is bounded. Then A has a fixed point. 3. Existence and uniqueness of solution After introducing those preliminary in Section 2, we obtain the existence result of the solution u of the variational formulation (2) via Lemma 1. Theorem 1. Under the assumptions (H1)–(H2). If f 2 L2 ðXÞ, then for all 2 6 p1 ; p2 < 1, there exists a solution u 2 V \ H 4 ðXÞ of the problem (2). Proof. For given u 2 V , from the embedding inequality and the assumption (H2), we have /ðjrujÞ 2 L2 ðXÞ
with
p
k/ðjrujÞkL2 6 j2 kukV2 :
Consider an auxiliary fourth-order linear problem ( Daðx; ru; DwÞ ¼ /ðjrujÞ þ f in X; w ¼ ow ¼0 on
on oX:
Associated variational formulation is Find w 2 V such that R R aðx; ru; DwÞDv dx ¼ X /ðjrujÞv þ fv dx X
8v 2 V :
Setting v = w in (4) gives Z m p p p j0 kwkV 6 j1 jruj 1 þ /ðjrujÞw þ fw dx 6 j1 kukV1 þ j2 kukV2 kwkV þ kf kL2 kwkV : X
ð4Þ
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Using Young’s inequality, one has p2 p1 1 kwkV 6 c3 kukVm þ kukVm1 þ kf km1 ; L2
ð5Þ
where c3 depends on j0 ; j1 ; j2 and m. By standard Galerkin approximation methods, there admits a weak solution w 2 V of (4). Since aðx; ru; DwÞ is linear with respect to Dw, then w is also a regular solution with kwkH 4 6 C;
ð6Þ
where C > 0 is independent of w. Denote the operator T : V ! V with T ðuÞ ¼ w defined by (4), then from (5) and (6), we have 8 p2 1 < kT ðuÞk 6 c kukpm1 þ kukm1 m1 3 V V þ kf kL2 V : kT ðuÞk 6 C:
ð7Þ
H4
In order to use Lemma 1, we have to show that T : V ! V is continuous and compact. Suppose uk ! u in V ;
ð8Þ
then according to (6), we have kwk kH 4 6 C: 1
Hence, there exists a subsequence fwkj gj¼1 V \ H 4 ðXÞ such that wk j ! w
in V
as j ! 1:
ð9Þ
For all v 2 V , Z Z aðx; ruk ; Dwkj ÞDv dx ¼ /ðjruk jÞv þ fv dx: X
X
In view of (8) and (9), Z Z aðx; ru; DwÞDv dx ¼ /ðjrujÞv þ fv dx X
X
as k; j ! þ1, which means that T ðuÞ ¼ w and gives the proof that T is continuous from V to V. Suppose fuk g V is a bounded sequence, then from (7), fT ðuk Þg is also a bounded sequence in V \ H 4 ðXÞ. Then there exists a subsequence fT ðukj Þg1 j¼1 with strong convergence in V. Hence T is also compact. Introduce the set D defined by D ¼ fu 2 V ; u ¼ kT ðuÞ for some 0 6 k 6 1g: Then for all u 2 D, we have uk ¼ T ðuÞ. Now we assume that there does not exist a k 2 ð0; 1Þ such that kukV is bounded. That is, if u 2 D, then kukV is unbounded for all k 2 ð0; 1Þ. Hence there exists a sufficiently large constant N > 0 such that kukV P N . We choose u0 2 D with ku0 k P N and km1 ¼ 0 j0 u0 < 1. Since ¼ T ðu Þ, then for all v 2 V , 0 p m p þ1m k0 2 j1 ku0 kV1 þj2 ku0 kV2 Z Z Du0 a x; ru0 ; /ðjru0 jÞv þ fv dx: Dv dx ¼ k0 X X Setting v ¼ uk00 , we obtain
m Z Z
u0
1 p1
6 j1 j0
jru j dx þ /ðjru0 jÞu0 þ fu0 dx: 0
k
k0 X k0 X 0 V That is
Z Z m p j0 ku0 kV 6 k0m1 j1 jru0 j 1 dx þ k0m1 /ðjru0 jÞu0 þ fu0 dx X X j0 p1 p2 þ1 m m1 6 k0 j1 ku0 kV þ j2 ku0 kV þ kf kL2 ku0 kV 6 ku0 kV þ kf kL2 ku0 kV : 2
R. An et al. / Applied Mathematics and Computation 194 (2007) 143–155
147
Therefore, ku0 kV 6
2 kf kL2 j0
m1 :
Thus, our assumption must be false and indeed for some 0 < k < 1, there exists a u 2 D such that kukV is bounded. So in view of Lemma 1, the variational problem (2) has a solution u 2 V \ H 4 ðXÞ. h If f 2 Lq ðXÞ; 2 6 q < þ1, then from argument of Theorem 1, we can get the higher regularity of the solution u. In fact, in view of Sobolev Embedding Theorem, for u 2 V , we also have /ðjrujÞ 2 Lq ðXÞ. Hence, we immediately derive the following corollary: Corollary 1. Under the assumptions of Theorem 1. If f 2 Lq ðXÞ; 2 6 q < þ1, then there exists a solution u 2 V \ W 4;q ðXÞ of the problem (2). For u 2 V , denote LðuÞ ¼ Daðx; ru; DuÞ /ðjrujÞ: Then the Frechet derivative of L at u is given by L0 ðuÞv ¼ DðaðuÞDv þ bðuÞ rvÞ /0 ðjrujÞv; where /0 ðjrujÞv is the Frechet derivative of / at u and bðuÞ ¼ ay ðx; ru; DuÞ 2 R2 :
aðuÞ ¼ az ðx; ru; DuÞ 2 R;
A fundamental assumption is (H4) aðwÞ P a0 for some constant a0 > 0 and all w 2 V . Denote /00 ðj ru jÞv v the second-order Frechet derivative of /. In terms of the assumption (H3) and Remark 1, we have Lemma 2. Given u; v 2 H 1 ðXÞ, then ( /0 ðjrujÞv 6 2j3 jrujp2 1 jrvj p2 2
00
/ ðjrujÞv v 6 2j4 jruj
a:e: in X;
jrvj
2
a:e: in X:
Proof. In view of (3), we have p 1
p 1
/ðjru þ trvjÞ /ðjrujÞ ðjru þ trvj 2 þ jruj 2 Þðjru þ trvj jrujÞ 6 j3 lim t!0 t!0 t t p2 1 p2 1 ðjru þ trvj þ jruj Þtjrvj p 1 ¼ 2j3 jruj 2 jrvj; a:e: in X; 6 j3 lim t!0 t
/0 ðjrujÞv ¼ lim
which gives the proof of the first estimate. Another can be showed with the similar method.
h
Introduce the convex set DðRÞ ¼ fu 2 V ; kukV 6 Rg; where R > 0. For simplicity, we denote C 1 ðRÞ ¼ sup jay ðx; ru; DuÞj ¼ sup jbðuÞj: u2DðRÞ
u2DðRÞ
Recall the proof of Theorem 1, the operator T ðuÞ ¼ w is defined by (4). Then we have following locally unique theorem Theorem 2. Suppose that there exists a sufficiently small R0 > 0 such that p 1
C 1 ðR0 Þ þ 2j3 R02
< a0 :
ð10Þ
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R. An et al. / Applied Mathematics and Computation 194 (2007) 143–155
If u1 ; u2 2 DðR0 Þ, then the operator T is a contraction operator such that p 1
kT ðu1 Þ T ðu2 ÞkV 6
C 1 ðR0 Þ þ 2j3 R02 a0
ku1 u2 kV :
Hence, the solution u of (2) is locally unique in DðR0 Þ. Proof. Let w1 ¼ T ðu1 Þ and w2 ¼ T ðu2 Þ, then Z ½/ðjru1 jÞ /ðjru2 jÞv dx X Z ¼ ½aðx; ru1 ; Dw1 Þ aðx; ru2 ; Dw2 ÞDv dx ZX ¼ ½aðx; ru1 ; Dw1 Þ aðx; ru1 ; Dw2 Þ þ aðx; ru1 ; Dw2 Þ aðx; ru2 ; Dw2 ÞDv dx ZX az jDw1 Dw2 jDv þ ay ðru1 ru2 ÞDv dx: ¼ X
Hence from Remark 1 and (H4), we have Z Z 2 p 1 p 1 jay kru1 ru2 kDw1 Dw2 jdx þ j3 ðjru1 j 2 þ jru2 j 2 Þjru1 ru2 kw1 w2 jdx a0 kw1 w2 kV 6 X
X
6 fC 1 ðR0 Þ þ
p 1 j3 ðku1 kV2
6 fC 1 ðR0 Þ þ
p 1 2j3 R02 gku1
þ
p 1 ku2 kV2 Þgku1
u2 kV kw1 w2 kV
u2 kV kw1 w2 kV :
Then p 1
kw1 w2 kV 6
C 1 ðR0 Þ þ 2j3 R02 a0
ku1 u2 kV :
4. Finite element approximation Let T h be a family of regular triangular (or quadrilateral) partition of X into triangles (or quadrilaterals) of diameter not greater than h. We define finite dimensional space ( fu 2 C 1 ðXÞ 8s 2 T h ; u 2 P 5 ðsÞg triangular partition; Wh ¼ fu 2 C 1 ðXÞ 8s 2 T h ; u 2 Q3 ðsÞg quadrilateral partition; where P 5 ðsÞ is the space of all polynomials defined on sP of degree less than or equal to 5, Q3 ðsÞ is the space of all polynomials in the reference space ðx; yÞ of the form cij xi y j , where the sum ranges over all integers i and j such that 0 6 i; j 6 3. Let V h ¼ W h \ V be the conforming finite element subspace of V. Then the discretized variational formulation of (2) is Find uh 2 V h such that R ð11Þ A1 ðuh ; vÞ ¼ X fv dx 8v 2 V h : According to the argument of Theorem 2, we also have the existence and uniqueness theorem of approximate solution uh of the discretized problem (11). Since the proof is almost the same as the proof of Theorem 2 by replacing u by uh, so we skip it here. Theorem 3. Under the assumptions of Theorem 2, there admits a unique discretized solution uh 2 DðR0 Þ of discretized variational problem (11).
R. An et al. / Applied Mathematics and Computation 194 (2007) 143–155
149
5. Error estimate Introduce the interpolation operator I h : V ! V h such that following interpolation approximation holds (cf. [18]): ku I h ukW j;p 6 j5 hkj kukW k;p
2 6 p 6 þ1; j ¼ 0; 1; 2; k ¼ 2; 3; 4
k;p
for u 2 W ðXÞ, where j5 > 0 is independent of h. Following assumption is also needed: (H5) kuh kL2 6 j6 h2 kuh kV for uh 2 V h , where j6 > 0 is independent of h. Introducing the bilinear form Z Z 0 A ðw; u; vÞ ¼ ðaðwÞDu þ bðwÞ ruÞDv dx /0 ðjrwjÞuv dx 8u; v 2 V ; X
X 2
where w 2 V . For w 2 V and g 2 H ðXÞ, we consider linear problem Find w 2 V such that A0 ðw; w; vÞ ¼ hg; vi 8v 2 V ;
ð12Þ
where h; i is the dual pairing between V and V 0 . Theorem 4. Under the assumption of Theorem 2. If w 2 DðR0 Þ, then there exists a unique solution w 2 V of (12) satisfying 1 kgkV 0 : kwkV 6 p 1 a0 C 1 ðR0 Þ 2j3 R02 Proof. Setting v ¼ w in (12) gives Z
2
aðwÞjDwj þ bðwÞ rwDw dx
Z
X
/0 ðjrwjÞww dx ¼ hg; wi:
X
Then from Lemma 2, one has Z 2 p 1 jbðwÞkrwkDwj þ 2j3 jrwj 2 jrwkwjdx þ kgkV 0 kwkV a0 kwkV 6 X p 1
p 1
2
2
6 ðC 1 ðR0 Þ þ 2j3 kwkV2 ÞkwkV þ kgkV 0 kwkV 6 ðC 1 ðR0 Þ þ 2j3 R02 ÞkwkV þ kgkV 0 kwkV : Hence, we have kwkV 6
1 p 1
a0 C 1 ðR0 Þ 2j3 R02
kgkV 0 :
By standard Galerkin approximation methods, we conclude that there exists a solution w 2 V . The proof of uniqueness is similar with Theorem 2. If w1 and w2 both are the solutions, then we can obtain 2
p 1
2
a0 kw1 w2 kV 6 ðC 1 ðR0 Þ þ 2j3 R02 Þkw1 w2 kV ; which gives w1 w2 . h By Theorem 4, the global inf–sup condition holds: sup v2V
A0 ðw; w; vÞ P j7 kwkV ; kvkV
where j7 > 0 maybe depends on w 2 DðR0 Þ. The discretized formulation associated with (12) is Find wh 2 V h such that A0 ðw; wh ; vÞ ¼ hg; vi 8v 2 V h :
ð13Þ
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R. An et al. / Applied Mathematics and Computation 194 (2007) 143–155
Similar with Theorem 4, we have Theorem 5. Under the assumption Theorem 2. If w 2 DðR0 Þ, then there exists a unique solution wh 2 V h of (13). By Theorem 5, the discretized inf–sup condition also holds: sup v2V h
A0 ðw; wh ; vÞ P j8 kwh kV ; kvkV
ð14Þ
where j8 > 0 may be depended on w 2 DðR0 Þ. Define the projection operator P : V ! V h by A0 ðw; Pu; vÞ ¼ A0 ðw; u; vÞ 8v 2 V h : Then (14) implies that kPukV 6 j9 kukV
8u 2 V ;
ð15Þ
where j9 > 0 is independent of h. From (15) we can show that Pu is a quasi-optimal approximation of u in V. Lemma 3 ku PukV 6 ðj9 þ 1Þ inf ku vkV : v2V h
ð16Þ
Proof. For any v 2 V h , according to (15), we have Pv ¼ v. Then ku PukV 6 ku vkV þ kPu vkV ¼ ku vkV þ kP ðu vÞkV 6 ð1 þ j9 Þku vkV ; h
which gives (16). Denote
gðtÞ ¼ A1 ðu þ tðuh uÞ; vÞ; then in view of 0
gð1Þ ¼ gð0Þ þ g ð0Þ þ
Z
1
g00 ðtÞð1 tÞdt; 0
we have A1 ðwh ; vÞ ¼ A1 ðw; vÞ þ A0 ðw; wh w; vÞ þ Rðw; wh ; vÞ; where Rðu; uh ; vÞ ¼
Z
1
g00 ðtÞð1 tÞdt: 0
Lemma 4. Suppose u and uh are the solutions of (2) and (11), respectively. Then A0 ðu; u uh ; vÞ ¼ Rðu; uh ; vÞ
8v 2 V h :
ð17Þ
jRðu; uh ; vÞj 6 C 2 ðMÞkDðu uh Þk2L4 kvkV
ð18Þ
Moreover, we have
for any given M > 0 such that kukW 2;1 þ kuh kW 2;1 6 M, where C 2 ðMÞ P
max ðj2ayz j þ jazz j þ jayy j þ j/00 jÞ:
jyj6M;jzj6M
Proof. (17) can be immediately obtained by subtracting (2) from (11). Next, we will show the estimate (18). Computing
R. An et al. / Applied Mathematics and Computation 194 (2007) 143–155
g00 ðtÞ ¼
Z
151
2
2ayz rðu uh ÞDðu uh ÞDv þ azz jDðu uh Þj Dv X
þ rðu uh Þ ayy rðu uh Þv /00 ðu uh Þ ðu uh Þv dx; where ayz 2 R2 ; ayy 2 R22 ; azz 2 R. Take C 2 ðMÞ P
max ðj2ayz j þ jazz j þ jayy j þ j/00 jÞ:
jyj6M;jzj6M
Then 2
2
2
jRðu; uh ; vÞj 6 C 2 ðMÞ½krðu uh ÞDðu uh ÞkL2 þ kDðu uh ÞkL4 þ krðu uh ÞkL4 kvkV 2
2
2
6 C 2 ðMÞ½kDðu uh ÞkL4 þ krðu uh ÞkL4 kvkV 6 C 2 ðMÞkDðu uh ÞkL4 kvkV :
Now setting v ¼ Pu uh in (17), one has A0 ðu; u uh ; Pu uh Þ ¼ A0 ðu; u Pu; Pu uh Þ þ A0 ðu; Pu uh ; Pu uh Þ ¼ A0 ðu; Pu uh ; Pu uh Þ ¼ Rðu; uh ; Pu uh Þ: From (14) and (18), we obtain 2
ku uh kV 6 ku PukV þ kPu uh kV 6 ð1 þ j9 Þ inf ku vkV þ j1 8 C 2 ðMÞkDðu uh ÞkL4 v2V h
6 ð1 þ j9 Þ inf ku vkV þ j1 8 C 2 ðMÞkDðu uh ÞkL1 ku uh kV v2V h
6 ð1 þ j9 Þ inf ku vkV þ j1 8 C 2 ðMÞkDðu I h uÞkL1 ku uh kV v2V h
þ
j1 8 C 2 ðMÞkDðuh
I h uÞkL1 ku uh kV
6 ð1 þ j9 Þ inf ku vkV þ j5 j1 8 C 2 ðMÞhkukW 3;1 ku uh kV v2V h
þ j1 8 C 2 ðMÞkDðuh I h uÞkL1 ku uh kV :
ð19Þ
Then for sufficiently small h, we have ku uh kV 6 ð1 þ j9 Þ inf ku vkV þ j1 8 C 2 ðMÞkDðuh I h uÞkL1 ku uh kV :
ð20Þ
v2V h
To discard the second the term of right-hand side of (20), we need following lemma. The idea of the proof comes from the methods of Green function introduced in [20]. Lemma 5. For each /h 2 V h , following inequality holds: kD/h kL1 6 j10 h1 k/h kV ;
ð21Þ
where j10 > 0 is independent of h. Proof. For any given z 2 X such that z is contained in some sz 2 T h . Denote Dirac function dz 2 C 1 0 ðsz Þ such that Z dz dx ¼ 1 jDk dz j 6 ch2k ; k ¼ 0; 1; 2; . . . ; sz
where c > 0 is independent of h and z. By construction DvðzÞ ¼ ðdz ; DvÞ
8v 2 V h :
Setting v ¼ /h gives Z 12 Z 12 Z 2 2 D/h ðzÞ 6 jdz kD/h jdx 6 jdz j dx jD/h j dx 6 j10 h1 k/h kV : sz
sz
X
152
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Setting /h ¼ uh I h u in (21) and substituting into (20), we obtain 1 ku uh kV 6 ð1 þ j9 Þ inf ku vkV þ j1 8 j10 C 2 ðMÞh kuh I h ukV ku uh kV v2V h
2
1 1 6 ð1 þ j9 Þ inf ku vkV þ j1 8 j10 C 2 ðMÞh ku uh kV þ j10 C 2 ðMÞh ku I h ukV ku uh kV v2V h
2
1 1 6 ð1 þ j9 Þ inf ku vkV þ j1 8 j10 C 2 ðMÞh ku uh kV þ j8 j10 C 2 ðMÞhkukH 4 ku uh kV : v2V h
Then for sufficiently small h, we have 2
1 ku uh kV 6 ð1 þ j9 Þ inf ku vkV þ j1 8 j10 C 2 ðMÞh ku uh kV : v2V h
Let 0 < h < 1. If 1 1 j1 8 j10 C 2 ðMÞh ku uh kV P h;
which also is 1
ku uh kV 6 ð1 hÞj8 j1 10 C 2 ðMÞ h; then ku uh kV 6 ð1 þ j9 Þh1 inf ku vkV 6 ch2 kukH 4 : v2V h
Summing up above, we obtain the H2-error estimate. Theorem 6. Suppose u and uh the solutions of (2) and (11) satisfying kukW 2;1 þ kuh kW 2;1 6 M: For sufficiently small h and u 2 W 4;2þe ðXÞ, where e is some positive constant, if u 2 DðR0 Þ and (10) holds, then we have ku uh kV 6 ch2 kukH 4 ; where c > 0 is independent of h. Remark 3. In Theorem 6, we require u 2 W 4;2þe ðXÞ for some e > 0. From Sobolev Embedding Theorem, then u 2 W 3;1 ðXÞ, which has been used in (19). If the solutions u and uh are sufficiently small, then from (17) and (19), we immediately obtain the following theorem. Theorem 7. Suppose u and uh the solutions of (2) and (11) satisfying e: kukW 2;1 þ kuh kW 2;1 6 M e is sufficiently small such that Suppose that u 2 DðR0 Þ. If M e ÞM e 6h j1 C 2 ð M 8
holds, then we have ku uh kV 6 ch2 kukH 4 ; where 0 < h < 1, c > 0 is independent of h. Proof. From (17) and (19), we have e e ku uh kV 6 ð1 þ j9 Þ inf ku vkV þ j1 8 C 2 ð M Þ M ku uh kV 6 ð1 þ j9 Þ inf ku vkV þ hku uh kV : v2V h
v2V h
Next, we prove L2-error estimate via duality technique. Consider the following biharmonic problem: 2 D w ¼ gðxÞ in X w ¼ ow ¼ 0 on oX: on
ð22Þ
R. An et al. / Applied Mathematics and Computation 194 (2007) 143–155
153
It is well known that kwkH 4 þ kwkV 6 ckgkL2 ; where c depends only on X. Setting w ¼ u uh , we have Z Z 2 ku uh kL2 ¼ ðu uh ; D2 wÞ ¼ Dðu uh ÞDðw wh Þdx þ Dðu uh ÞDwh dx X X Z Z 1 1 6 ku uh kV kw wh kV þ ð/ðjrujÞ /ðjruh jÞÞ/h dx ay rðu uh ÞDwh dx a X a X Z p 1 p 1 6 ku uh kV kw wh kV þ c ðjruj 2 þ jruh j 2 Þjru ruh k/h jdx X Z þ c jrðu uh ÞkDwh jdx X p 1
p 1
6 ch4 ku uh kL2 kukH 4 þ cðkukW2 1;1 þ kuh kW2 1;1 Þku uh kV kwh kL2 þ ckrðu uh ÞkL2 kwh kV 6 ch4 ku uh kL2 kukH 4 þ cM p2 1 h2 ku uh kV kwh kV þ krðu I h uÞkL2 kwh kV þ krðI h u uh ÞkL2 kwh kV 6 ch4 ku uh kL2 kukH 4 þ ch3 ku uh kL2 kukH 4 þ chku uh kL2 kuh I h ukV 6 ch3 ku uh kL2 kukH 4 þ chku uh kL2 kuh ukV þ chku uh kL2 ku I h ukV 6 ch3 ku uh kL2 kukH 4 : Summing up above, we obtain the L2-error estimate. Theorem 8. Under the assumptions of Theorem 6, for sufficiently small h, following L2-error estimate holds: ku uh kL2 6 ch3 kukH 4 ;
ð23Þ
where c > 0 is independent of h. Remark 4. The L2-error estimate (22) is not optimal. However, from the numerical experiments in Section 6, the convergence rate of L2-error is Oðh4 Þ if h is sufficiently small. In addition, if ay 0, then (22) can be improved to be optimal. Corollary 2. Under the assumptions of Theorem 8. If ay 0, then we have ku uh kL2 6 ch4 kukH 4 ; where c > 0 is independent of h. 6. Numerical results The numerical experiments in this section are designed to verity the error estimate obtained in Section 5. Here, we use Bogner–Fox–Schmidt element (Q3 Element). Consider the following fourth-order nonlinear problem: ( 2 4 DðDu þ 2jruj Þ ¼ jruj þ f ; in X; ð24Þ u ¼ ou ¼0 on oX; on where X ¼ fðx; yÞ; 0 6 x 6 1; 0 6 y 6 1g. Choose a true solution 2
2
u ¼ sinðpxÞ sinðpyÞ : It is easily to verify the u satisfying the homogeneous Dirichlet boundary conditions. Select u0 2 V satisfying ðDu0 ; DvÞ ¼ ðf ; vÞ
8v 2 V
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R. An et al. / Applied Mathematics and Computation 194 (2007) 143–155
as the initial value for iteration procedures. If un is known, then we compute unþ1 by the following simple iteration: ðDunþ1 ; DvÞ ¼ ðjrun j4 þ f ; vÞ ð2jrun j2 ; DvÞ
8v 2 V :
Denote M and N be the division numbers at x-axis and y-axis, respectively. Let K be the iteration numbers. Tables 1 and 2 display that the convergence rate of H2-error is almost Oðh2 Þ, which is consistent with Theorem 6. From Tables 3 and 4, we conclude that the optimal convergence order of L2-error on the problem (24) is obtained only for sufficiently small h. Table 1 The H2 relative error when M = N = H and K = 25 H
kuKh uK1 kV h kuK1 kV h
kuuh kV kukV
Order
8 16 32 64
0.997373244756572 0.998345931123661 0.998873146002898 0.999334728029242
7.967491672694477E002 1.994969017256056E002 4.997594771356409E003 1.313073616149285E003
n 2.00 2.00 1.93
Table 2 The H2 relative error when M = N = H and K = 30 H
kuKh uK1 kV h kuK1 kV h
kuuh kV kukV
Order
8 16 32 64
0.997372584913545 0.998344880491724 0.998874169146644 0.999336347862812
7.964694564277870E002 1.994549096190196E002 4.982081713114712E003 1.251366164489329E003
n 2.00 2.00 1.99
Table 3 The L2 relative error when M = N = H and K = 50 and ay 6¼ 0 H
kuKh uK1 kL2 h kuK1 kL2 h
kuuh kL2 kukL2
Order
8 16 32 64
0.990658875734683 0.977849077497979 0.930234820307248 0.807448430951118
8.773064381523671E003 8.832622608842352E004 5.230972374381978E005 3.245313951806216E006
n 3.31 4.08 4.01
Table 4 The L2 relative error when M = N = H and K = 60 and ay 6¼ 0 H
kuKh uK1 kL2 h kuK1 kL2 h
kuuh kL2 kukL2
Order
8 16 32 64
0.990658875111902 0.977849092199866 0.930234868567753 0.807448512100228
8.773066096053444E003 8.835136311855687E004 5.245861350625196E005 3.207989157622057E006
n 3.31 4.07 4.03
Table 5 The L2 relative error when M = N = H and K = 12 and ay 0 H
kuKh uK1 kL2 h kuK1 kL2 h
kuuh kL2 kukL2
Order
8 16 32 64
0.990624878485068 0.977799305369950 0.930212379795423 0.807440047682768
1.928800836470502E002 9.519844978571929E004 6.063129261860954E005 4.400933964318014E006
n 4.34 3.97 3.78
R. An et al. / Applied Mathematics and Computation 194 (2007) 143–155
155
Table 6 The L2 relative error when M = N = H and K = 15 and ay 0 H
kuKh uK1 kL2 h kuK1 kL2 h
kuuh kL2 kukL2
Order
8 16 32 64
0.990624878340708 0.977799305825940 0.930212385446254 0.807440138810543
1.928734477005187E002 9.513391037479355E004 5.998463989670770E005 3.737223748694858E006
n 4.34 3.99 4.00
Finally, we give a numerical experiment when ay 0. Consider ( 4 D2 u ¼ jruj þ f ; in X; u ¼ ou ¼0 on oX: on
ð25Þ
Tables 5 and 6 tell us that if ay 0, the optimal convergence order of L2-error is obtained. Acknowledgements Supported by the Nature Science Foundation of China (Grant Nos. 50306019, 10571142, 10471110 and 10471109). References [1] O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, 1969. [2] L.A. Peletier, W.C. Troy, Spatial patterns. Higher order models in physics and mechanics, Progress in Nonlinear Differerential Equations Applications, vol. 45, Birkhauser, Boston, MA, 2001. [3] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959) 623–727. [4] E. Berchio, F. Gazzola, Some remarks on biharmonic elliptic problems with positive, increasing and convex nonlinearities, Electron. J. Differ. Equat. 2005 (34) (2005) 1–20. [5] G. Arioli, F. Gazzola, H.C. Grunau, E. Mitidieri, A semilinear fourth order elliptic problem with exponential nonlinearity, SIAM J. Math. Anal. 36 (4) (2005) 1226–1258. [6] G. Arioli, F. Gazzola, H.C. Grunau, Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity, J. Differ. Equat. 230 (2006) 743–770. [7] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1998. [8] L.C. Evans, Partial Differential Equations, Providence American Mathematical Society, 1998. [9] Kaitai Li, Jian Su, Limin Gao, Optimal shape design for blade’s surface of an impeller via the Navier–Stokes equations, Commun. Numer. Meth. Eng. 22 (6) (2006) 657–676. [10] F. Brezzi, P. Raviart, Mixed finite element methods for fourth order elliptic equations, in: J. Miller (Ed.), Topic in Numerical Analysis III, Academic Press, New York, 1978. [11] P. Ciarlet, P. Raviart, A mixed finite element method for the biharmonic equation, in: C. de Boor (Ed.), Symposium on Mathematical Aspects of Finite Element in Partial Differential Equations, Academic Press, New York, 1974, pp. 125–143. [12] L. Herrmann, Finite element bending analysis for plates, J. Eng. Mech., Div. ASCE EM5 93 (1967) 49–83. [13] T. Miyoshi, A finite element method for the solution of fourth order partial differential equations, Kunamoto J. Sci. (Math.) 9 (1973) 87–116. [14] C. Johnson, On the convergence of a mixed finite element method for plate bending problems, Numer. Math. 21 (1973) 43–62. [15] I. Babuska, J. Osborn, J. Pitaranta, Analysis of mixed methods using mesh dependent norms, Math. Com. 35 (152) (1980) 1039–1062. [16] M.M. Karchevskii, A.D. Lyashko, M.R. Timerbaev, A mixed finite-element method for quasilinear degenerate fourth-order elliptic equations, Differential Equations 36 (7) (2000) 1050–1057. [17] M.M. Karchevskii, A.D. Lyashko, M.R. Timerbaev, A finite element method for fourth-order quasilinear degenerating elliptic equations, Differential Equations 35 (2) (1999) 233–238. [18] Zhangxing Chen, Finite Element Methods and Their Application, Spring-Verlag, 2005. [19] Jinchao Xu, Two-grid discretization techniques for linear and nonlinear PDE, SIAM J. Numer. Anal. 33 (1996) 1759–1777. [20] R. Rannacher, R. Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comput. 38 (1982) 437–445.
Applied Mathematics and Computation 194 (2007) 143–155 www.elsevier.com/locate/amc
Finite element approximation for fourth-order nonlinear problem in the plane Rong An *, Yuan Li, Kaitai Li Department of Mathematics, Xi’an Jiaotong University, Xi’an 710049, China
Abstract In this paper, a class of fourth-order nonlinear elliptic problem is investigated in a bounded convex domain X R2 . Under some assumptions, the existence and uniqueness of solution are proved via the Schaefer’s Fixed Point Theorem. Furthermore, conforming finite element approximation is applied and H2-error estimate and L2-error estimate are obtained. Finally, the numerical experiments are provided to verify our theoretical analysis. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Fourth-order nonlinear problem; Finite element approximation; Error estimates
1. Introduction The fourth-order elliptic problem is an important mathematical model in fluid mechanics [1], solid mechanics and other continuous physical system [2]. For example, in fluid mechanics, the stream function w of the incompressible flows in R2 satisfies the biharmonic problem; in solid mechanics, plate bending model is satisfied by harmonic problem, etc. Hence, large numbers of papers appear about this problem. As well known, the well-posedness theory of fourth-order linear problem is similar with second-order-problem [3]. Although many techniques like the maximum principle and comparison principle are not available for fourth-order problem with Dirichlet boundary, others still are valid, such as the methods of upper and lower solutions, and minimax method, in particular, if the equation is a Euler–Lagrange equation associated with some energy functional [4–6]. In addition, some fixed point theorems also are power tools for dealing with the nonlinear problem. We refer the reader to [7,8] for more details. In this paper, we will study a class of fourth-order nonlinear problem: Daðx; ru; DuÞ ¼ /ðjrujÞ þ f which comes from the mathematical model of the optimal shape design for blade’s surface of an impeller, which is a coupled system with nonlinear fourth-order elliptic problem and rotating Navier–Stokes equations *
Corresponding author. E-mail address: [email protected] (R. An).
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.04.060
144
R. An et al. / Applied Mathematics and Computation 194 (2007) 143–155
[9]. Under some assumptions, the Schaefer’s Fixed Point Theorem is applied to prove the existence of solution. Comparing with other fixed point theorems, the advantage of Contracting Mapping Principle is that the solution not only exists but also is unique. Hence, in terms of Contracting Mapping Principle, we show that the solution is locally unique. In numerical analysis, the popular method for fourth-order problem is mixed finite element methods, such as [10–16] and references therein. Moreover, to our best knowledge, [16,17] are only two papers for fourth-order quasilinear problem. In [17], Karchevskii, Lyashko and Timerbaev discussed conforming finite element. But very strong assumptions were required. They assumed that the nonlinear operator of the problem was strongly monotone and Lipschitz continuous, from which the existence and uniqueness of weak solution and approximation solution were immediately obtained in an appropriate weighted Sobolev space. In this paper, we also investigate the finite element approximation using conforming element under some weak assumptions. It is well known that for linear biharmonic problem following error estimates hold (cf. [18]): ku uh kH 2 6 chðkukH 3 þ hkukH 4 Þ; ku uh kL2 6 ch3 ðkukH 3 þ hkukH 4 Þ; where c > 0 is independent of h. In this paper, the H2-error estimate is showed to be optimal via a linearized technique which is introduced and used to deal with the second-order nonlinear problem in [19]. However, the L2-error estimate is not optimal unless ay i ðx; y; zÞ 0, where y 2 R2 ; z 2 R; i ¼ 1; 2. This paper is organized as follows: in Section 2, we give some preliminary knowledge; in Section 3, the existence and uniqueness of solution are showed; in Section 4, the finite element subspace and the discretized variational formulation are given. Like Section 3, the approximation solution is locally unique; in Section 5, the H2-error estimate and L2-error estimate for sufficiently small h are obtained; in Section 6, the numerical experiments are provided to verify our theoretical analysis. 2. Preliminary Consider a class of fourth-order nonlinear problem: ( Daðx; ru; DuÞ ¼ /ðjrujÞ þ f in X; u ¼ ou ¼0 on
on oX;
ð1Þ
where uðxÞ : X ! R is unknown, aðx; y; zÞ : X R2 R ! R is a smooth function. Moreover, að; y; zÞ is nonlinear with respect to y and is linear with respect to z, /ðÞ : R ! R is a nonlinear operator, f is independent of u. X R2 is a bounded convex domain with boundary oX, n denotes the unite outer normal vector to oX. Introduce the linear space ov 2 2 ¼ 0 on oX V ¼ H 0 ðXÞ ¼ v 2 H ðXÞ; v ¼ on with the inner product Z Aðu; vÞ ¼ DuDv dx
8u; v 2 V
X
and the norm 1
kvkV ¼ Aðv; vÞ2
8v 2 V :
Following Sobolev inequalities are usually used: kukL1 6 c1 kukV ;
krukLq 6 c2 kukV
for all 2 6 q < þ1; 8u 2 V ;
where c1 and c2 depend only on X. Throughout this paper, we always assume the embedding constant c1 ¼ c2 ¼ 1.
R. An et al. / Applied Mathematics and Computation 194 (2007) 143–155
For f 2 V 0 ¼ H 2 ðXÞ, the variational formulation associated with the problem (1) is Find u 2 V such that R A1 ðu; vÞ ¼ X fv dx 8v 2 V ; where A1 ðu; vÞ ¼
Z
145
ð2Þ
aðx; ru; DuÞDv /ðjrujÞv dx:
X
Suppose that the nonlinear term aðx; y; zÞ and /ðsÞ satisfy m
p
(H1) aðx; y; zÞz P j0 jzj j1 jyj 1 , where m P 2, p1 > m, j0 and j1 both are positive constants and are independent of y; z and p1; p (H2) j/ðsÞj 6 j2 jsj 2 , where 2 6 p2 < 1; j2 is a positive constant and is independent of s and p2; p 1 p 2 2 (H3) / 2 C and j/0 ðsÞj 6 j3 jsj 2 , j/00 ðsÞj 6 j4 jsj 2 , where j3 and j4 both are positive constants and are independent of s and p2. Remark 1. From (H3), in view of mean-value theorem, we have j/ðs1 Þ /ðs2 Þj 6 j3 jas1 þ ð1 aÞs2 j
p2 1
p2 1
js1 s2 j 6 j3 ðjs1 j
p2 1
þ js2 j
Þjs1 s2 j;
ð3Þ
where 0 6 a 6 1. Remark 2. The condition p1 > m in (H1) can be replaced by p2 þ 1 > m in (H2). Following Schaefer’s Fixed Point Theorem plays a key role in the existence of solution. Lemma 1 [2, Theorem 9.4]. Let X be a real Banach space. Suppose that A : X ! X is a continuous and compact mapping. Assume further that the set fu 2 X ju ¼ kAðuÞ for some 0 6 k 6 1g is bounded. Then A has a fixed point. 3. Existence and uniqueness of solution After introducing those preliminary in Section 2, we obtain the existence result of the solution u of the variational formulation (2) via Lemma 1. Theorem 1. Under the assumptions (H1)–(H2). If f 2 L2 ðXÞ, then for all 2 6 p1 ; p2 < 1, there exists a solution u 2 V \ H 4 ðXÞ of the problem (2). Proof. For given u 2 V , from the embedding inequality and the assumption (H2), we have /ðjrujÞ 2 L2 ðXÞ
with
p
k/ðjrujÞkL2 6 j2 kukV2 :
Consider an auxiliary fourth-order linear problem ( Daðx; ru; DwÞ ¼ /ðjrujÞ þ f in X; w ¼ ow ¼0 on
on oX:
Associated variational formulation is Find w 2 V such that R R aðx; ru; DwÞDv dx ¼ X /ðjrujÞv þ fv dx X
8v 2 V :
Setting v = w in (4) gives Z m p p p j0 kwkV 6 j1 jruj 1 þ /ðjrujÞw þ fw dx 6 j1 kukV1 þ j2 kukV2 kwkV þ kf kL2 kwkV : X
ð4Þ
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R. An et al. / Applied Mathematics and Computation 194 (2007) 143–155
Using Young’s inequality, one has p2 p1 1 kwkV 6 c3 kukVm þ kukVm1 þ kf km1 ; L2
ð5Þ
where c3 depends on j0 ; j1 ; j2 and m. By standard Galerkin approximation methods, there admits a weak solution w 2 V of (4). Since aðx; ru; DwÞ is linear with respect to Dw, then w is also a regular solution with kwkH 4 6 C;
ð6Þ
where C > 0 is independent of w. Denote the operator T : V ! V with T ðuÞ ¼ w defined by (4), then from (5) and (6), we have 8 p2 1 < kT ðuÞk 6 c kukpm1 þ kukm1 m1 3 V V þ kf kL2 V : kT ðuÞk 6 C:
ð7Þ
H4
In order to use Lemma 1, we have to show that T : V ! V is continuous and compact. Suppose uk ! u in V ;
ð8Þ
then according to (6), we have kwk kH 4 6 C: 1
Hence, there exists a subsequence fwkj gj¼1 V \ H 4 ðXÞ such that wk j ! w
in V
as j ! 1:
ð9Þ
For all v 2 V , Z Z aðx; ruk ; Dwkj ÞDv dx ¼ /ðjruk jÞv þ fv dx: X
X
In view of (8) and (9), Z Z aðx; ru; DwÞDv dx ¼ /ðjrujÞv þ fv dx X
X
as k; j ! þ1, which means that T ðuÞ ¼ w and gives the proof that T is continuous from V to V. Suppose fuk g V is a bounded sequence, then from (7), fT ðuk Þg is also a bounded sequence in V \ H 4 ðXÞ. Then there exists a subsequence fT ðukj Þg1 j¼1 with strong convergence in V. Hence T is also compact. Introduce the set D defined by D ¼ fu 2 V ; u ¼ kT ðuÞ for some 0 6 k 6 1g: Then for all u 2 D, we have uk ¼ T ðuÞ. Now we assume that there does not exist a k 2 ð0; 1Þ such that kukV is bounded. That is, if u 2 D, then kukV is unbounded for all k 2 ð0; 1Þ. Hence there exists a sufficiently large constant N > 0 such that kukV P N . We choose u0 2 D with ku0 k P N and km1 ¼ 0 j0 u0 < 1. Since ¼ T ðu Þ, then for all v 2 V , 0 p m p þ1m k0 2 j1 ku0 kV1 þj2 ku0 kV2 Z Z Du0 a x; ru0 ; /ðjru0 jÞv þ fv dx: Dv dx ¼ k0 X X Setting v ¼ uk00 , we obtain
m Z Z
u0
1 p1
6 j1 j0
jru j dx þ /ðjru0 jÞu0 þ fu0 dx: 0
k
k0 X k0 X 0 V That is
Z Z m p j0 ku0 kV 6 k0m1 j1 jru0 j 1 dx þ k0m1 /ðjru0 jÞu0 þ fu0 dx X X j0 p1 p2 þ1 m m1 6 k0 j1 ku0 kV þ j2 ku0 kV þ kf kL2 ku0 kV 6 ku0 kV þ kf kL2 ku0 kV : 2
R. An et al. / Applied Mathematics and Computation 194 (2007) 143–155
147
Therefore, ku0 kV 6
2 kf kL2 j0
m1 :
Thus, our assumption must be false and indeed for some 0 < k < 1, there exists a u 2 D such that kukV is bounded. So in view of Lemma 1, the variational problem (2) has a solution u 2 V \ H 4 ðXÞ. h If f 2 Lq ðXÞ; 2 6 q < þ1, then from argument of Theorem 1, we can get the higher regularity of the solution u. In fact, in view of Sobolev Embedding Theorem, for u 2 V , we also have /ðjrujÞ 2 Lq ðXÞ. Hence, we immediately derive the following corollary: Corollary 1. Under the assumptions of Theorem 1. If f 2 Lq ðXÞ; 2 6 q < þ1, then there exists a solution u 2 V \ W 4;q ðXÞ of the problem (2). For u 2 V , denote LðuÞ ¼ Daðx; ru; DuÞ /ðjrujÞ: Then the Frechet derivative of L at u is given by L0 ðuÞv ¼ DðaðuÞDv þ bðuÞ rvÞ /0 ðjrujÞv; where /0 ðjrujÞv is the Frechet derivative of / at u and bðuÞ ¼ ay ðx; ru; DuÞ 2 R2 :
aðuÞ ¼ az ðx; ru; DuÞ 2 R;
A fundamental assumption is (H4) aðwÞ P a0 for some constant a0 > 0 and all w 2 V . Denote /00 ðj ru jÞv v the second-order Frechet derivative of /. In terms of the assumption (H3) and Remark 1, we have Lemma 2. Given u; v 2 H 1 ðXÞ, then ( /0 ðjrujÞv 6 2j3 jrujp2 1 jrvj p2 2
00
/ ðjrujÞv v 6 2j4 jruj
a:e: in X;
jrvj
2
a:e: in X:
Proof. In view of (3), we have p 1
p 1
/ðjru þ trvjÞ /ðjrujÞ ðjru þ trvj 2 þ jruj 2 Þðjru þ trvj jrujÞ 6 j3 lim t!0 t!0 t t p2 1 p2 1 ðjru þ trvj þ jruj Þtjrvj p 1 ¼ 2j3 jruj 2 jrvj; a:e: in X; 6 j3 lim t!0 t
/0 ðjrujÞv ¼ lim
which gives the proof of the first estimate. Another can be showed with the similar method.
h
Introduce the convex set DðRÞ ¼ fu 2 V ; kukV 6 Rg; where R > 0. For simplicity, we denote C 1 ðRÞ ¼ sup jay ðx; ru; DuÞj ¼ sup jbðuÞj: u2DðRÞ
u2DðRÞ
Recall the proof of Theorem 1, the operator T ðuÞ ¼ w is defined by (4). Then we have following locally unique theorem Theorem 2. Suppose that there exists a sufficiently small R0 > 0 such that p 1
C 1 ðR0 Þ þ 2j3 R02
< a0 :
ð10Þ
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R. An et al. / Applied Mathematics and Computation 194 (2007) 143–155
If u1 ; u2 2 DðR0 Þ, then the operator T is a contraction operator such that p 1
kT ðu1 Þ T ðu2 ÞkV 6
C 1 ðR0 Þ þ 2j3 R02 a0
ku1 u2 kV :
Hence, the solution u of (2) is locally unique in DðR0 Þ. Proof. Let w1 ¼ T ðu1 Þ and w2 ¼ T ðu2 Þ, then Z ½/ðjru1 jÞ /ðjru2 jÞv dx X Z ¼ ½aðx; ru1 ; Dw1 Þ aðx; ru2 ; Dw2 ÞDv dx ZX ¼ ½aðx; ru1 ; Dw1 Þ aðx; ru1 ; Dw2 Þ þ aðx; ru1 ; Dw2 Þ aðx; ru2 ; Dw2 ÞDv dx ZX az jDw1 Dw2 jDv þ ay ðru1 ru2 ÞDv dx: ¼ X
Hence from Remark 1 and (H4), we have Z Z 2 p 1 p 1 jay kru1 ru2 kDw1 Dw2 jdx þ j3 ðjru1 j 2 þ jru2 j 2 Þjru1 ru2 kw1 w2 jdx a0 kw1 w2 kV 6 X
X
6 fC 1 ðR0 Þ þ
p 1 j3 ðku1 kV2
6 fC 1 ðR0 Þ þ
p 1 2j3 R02 gku1
þ
p 1 ku2 kV2 Þgku1
u2 kV kw1 w2 kV
u2 kV kw1 w2 kV :
Then p 1
kw1 w2 kV 6
C 1 ðR0 Þ þ 2j3 R02 a0
ku1 u2 kV :
4. Finite element approximation Let T h be a family of regular triangular (or quadrilateral) partition of X into triangles (or quadrilaterals) of diameter not greater than h. We define finite dimensional space ( fu 2 C 1 ðXÞ 8s 2 T h ; u 2 P 5 ðsÞg triangular partition; Wh ¼ fu 2 C 1 ðXÞ 8s 2 T h ; u 2 Q3 ðsÞg quadrilateral partition; where P 5 ðsÞ is the space of all polynomials defined on sP of degree less than or equal to 5, Q3 ðsÞ is the space of all polynomials in the reference space ðx; yÞ of the form cij xi y j , where the sum ranges over all integers i and j such that 0 6 i; j 6 3. Let V h ¼ W h \ V be the conforming finite element subspace of V. Then the discretized variational formulation of (2) is Find uh 2 V h such that R ð11Þ A1 ðuh ; vÞ ¼ X fv dx 8v 2 V h : According to the argument of Theorem 2, we also have the existence and uniqueness theorem of approximate solution uh of the discretized problem (11). Since the proof is almost the same as the proof of Theorem 2 by replacing u by uh, so we skip it here. Theorem 3. Under the assumptions of Theorem 2, there admits a unique discretized solution uh 2 DðR0 Þ of discretized variational problem (11).
R. An et al. / Applied Mathematics and Computation 194 (2007) 143–155
149
5. Error estimate Introduce the interpolation operator I h : V ! V h such that following interpolation approximation holds (cf. [18]): ku I h ukW j;p 6 j5 hkj kukW k;p
2 6 p 6 þ1; j ¼ 0; 1; 2; k ¼ 2; 3; 4
k;p
for u 2 W ðXÞ, where j5 > 0 is independent of h. Following assumption is also needed: (H5) kuh kL2 6 j6 h2 kuh kV for uh 2 V h , where j6 > 0 is independent of h. Introducing the bilinear form Z Z 0 A ðw; u; vÞ ¼ ðaðwÞDu þ bðwÞ ruÞDv dx /0 ðjrwjÞuv dx 8u; v 2 V ; X
X 2
where w 2 V . For w 2 V and g 2 H ðXÞ, we consider linear problem Find w 2 V such that A0 ðw; w; vÞ ¼ hg; vi 8v 2 V ;
ð12Þ
where h; i is the dual pairing between V and V 0 . Theorem 4. Under the assumption of Theorem 2. If w 2 DðR0 Þ, then there exists a unique solution w 2 V of (12) satisfying 1 kgkV 0 : kwkV 6 p 1 a0 C 1 ðR0 Þ 2j3 R02 Proof. Setting v ¼ w in (12) gives Z
2
aðwÞjDwj þ bðwÞ rwDw dx
Z
X
/0 ðjrwjÞww dx ¼ hg; wi:
X
Then from Lemma 2, one has Z 2 p 1 jbðwÞkrwkDwj þ 2j3 jrwj 2 jrwkwjdx þ kgkV 0 kwkV a0 kwkV 6 X p 1
p 1
2
2
6 ðC 1 ðR0 Þ þ 2j3 kwkV2 ÞkwkV þ kgkV 0 kwkV 6 ðC 1 ðR0 Þ þ 2j3 R02 ÞkwkV þ kgkV 0 kwkV : Hence, we have kwkV 6
1 p 1
a0 C 1 ðR0 Þ 2j3 R02
kgkV 0 :
By standard Galerkin approximation methods, we conclude that there exists a solution w 2 V . The proof of uniqueness is similar with Theorem 2. If w1 and w2 both are the solutions, then we can obtain 2
p 1
2
a0 kw1 w2 kV 6 ðC 1 ðR0 Þ þ 2j3 R02 Þkw1 w2 kV ; which gives w1 w2 . h By Theorem 4, the global inf–sup condition holds: sup v2V
A0 ðw; w; vÞ P j7 kwkV ; kvkV
where j7 > 0 maybe depends on w 2 DðR0 Þ. The discretized formulation associated with (12) is Find wh 2 V h such that A0 ðw; wh ; vÞ ¼ hg; vi 8v 2 V h :
ð13Þ
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R. An et al. / Applied Mathematics and Computation 194 (2007) 143–155
Similar with Theorem 4, we have Theorem 5. Under the assumption Theorem 2. If w 2 DðR0 Þ, then there exists a unique solution wh 2 V h of (13). By Theorem 5, the discretized inf–sup condition also holds: sup v2V h
A0 ðw; wh ; vÞ P j8 kwh kV ; kvkV
ð14Þ
where j8 > 0 may be depended on w 2 DðR0 Þ. Define the projection operator P : V ! V h by A0 ðw; Pu; vÞ ¼ A0 ðw; u; vÞ 8v 2 V h : Then (14) implies that kPukV 6 j9 kukV
8u 2 V ;
ð15Þ
where j9 > 0 is independent of h. From (15) we can show that Pu is a quasi-optimal approximation of u in V. Lemma 3 ku PukV 6 ðj9 þ 1Þ inf ku vkV : v2V h
ð16Þ
Proof. For any v 2 V h , according to (15), we have Pv ¼ v. Then ku PukV 6 ku vkV þ kPu vkV ¼ ku vkV þ kP ðu vÞkV 6 ð1 þ j9 Þku vkV ; h
which gives (16). Denote
gðtÞ ¼ A1 ðu þ tðuh uÞ; vÞ; then in view of 0
gð1Þ ¼ gð0Þ þ g ð0Þ þ
Z
1
g00 ðtÞð1 tÞdt; 0
we have A1 ðwh ; vÞ ¼ A1 ðw; vÞ þ A0 ðw; wh w; vÞ þ Rðw; wh ; vÞ; where Rðu; uh ; vÞ ¼
Z
1
g00 ðtÞð1 tÞdt: 0
Lemma 4. Suppose u and uh are the solutions of (2) and (11), respectively. Then A0 ðu; u uh ; vÞ ¼ Rðu; uh ; vÞ
8v 2 V h :
ð17Þ
jRðu; uh ; vÞj 6 C 2 ðMÞkDðu uh Þk2L4 kvkV
ð18Þ
Moreover, we have
for any given M > 0 such that kukW 2;1 þ kuh kW 2;1 6 M, where C 2 ðMÞ P
max ðj2ayz j þ jazz j þ jayy j þ j/00 jÞ:
jyj6M;jzj6M
Proof. (17) can be immediately obtained by subtracting (2) from (11). Next, we will show the estimate (18). Computing
R. An et al. / Applied Mathematics and Computation 194 (2007) 143–155
g00 ðtÞ ¼
Z
151
2
2ayz rðu uh ÞDðu uh ÞDv þ azz jDðu uh Þj Dv X
þ rðu uh Þ ayy rðu uh Þv /00 ðu uh Þ ðu uh Þv dx; where ayz 2 R2 ; ayy 2 R22 ; azz 2 R. Take C 2 ðMÞ P
max ðj2ayz j þ jazz j þ jayy j þ j/00 jÞ:
jyj6M;jzj6M
Then 2
2
2
jRðu; uh ; vÞj 6 C 2 ðMÞ½krðu uh ÞDðu uh ÞkL2 þ kDðu uh ÞkL4 þ krðu uh ÞkL4 kvkV 2
2
2
6 C 2 ðMÞ½kDðu uh ÞkL4 þ krðu uh ÞkL4 kvkV 6 C 2 ðMÞkDðu uh ÞkL4 kvkV :
Now setting v ¼ Pu uh in (17), one has A0 ðu; u uh ; Pu uh Þ ¼ A0 ðu; u Pu; Pu uh Þ þ A0 ðu; Pu uh ; Pu uh Þ ¼ A0 ðu; Pu uh ; Pu uh Þ ¼ Rðu; uh ; Pu uh Þ: From (14) and (18), we obtain 2
ku uh kV 6 ku PukV þ kPu uh kV 6 ð1 þ j9 Þ inf ku vkV þ j1 8 C 2 ðMÞkDðu uh ÞkL4 v2V h
6 ð1 þ j9 Þ inf ku vkV þ j1 8 C 2 ðMÞkDðu uh ÞkL1 ku uh kV v2V h
6 ð1 þ j9 Þ inf ku vkV þ j1 8 C 2 ðMÞkDðu I h uÞkL1 ku uh kV v2V h
þ
j1 8 C 2 ðMÞkDðuh
I h uÞkL1 ku uh kV
6 ð1 þ j9 Þ inf ku vkV þ j5 j1 8 C 2 ðMÞhkukW 3;1 ku uh kV v2V h
þ j1 8 C 2 ðMÞkDðuh I h uÞkL1 ku uh kV :
ð19Þ
Then for sufficiently small h, we have ku uh kV 6 ð1 þ j9 Þ inf ku vkV þ j1 8 C 2 ðMÞkDðuh I h uÞkL1 ku uh kV :
ð20Þ
v2V h
To discard the second the term of right-hand side of (20), we need following lemma. The idea of the proof comes from the methods of Green function introduced in [20]. Lemma 5. For each /h 2 V h , following inequality holds: kD/h kL1 6 j10 h1 k/h kV ;
ð21Þ
where j10 > 0 is independent of h. Proof. For any given z 2 X such that z is contained in some sz 2 T h . Denote Dirac function dz 2 C 1 0 ðsz Þ such that Z dz dx ¼ 1 jDk dz j 6 ch2k ; k ¼ 0; 1; 2; . . . ; sz
where c > 0 is independent of h and z. By construction DvðzÞ ¼ ðdz ; DvÞ
8v 2 V h :
Setting v ¼ /h gives Z 12 Z 12 Z 2 2 D/h ðzÞ 6 jdz kD/h jdx 6 jdz j dx jD/h j dx 6 j10 h1 k/h kV : sz
sz
X
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R. An et al. / Applied Mathematics and Computation 194 (2007) 143–155
Setting /h ¼ uh I h u in (21) and substituting into (20), we obtain 1 ku uh kV 6 ð1 þ j9 Þ inf ku vkV þ j1 8 j10 C 2 ðMÞh kuh I h ukV ku uh kV v2V h
2
1 1 6 ð1 þ j9 Þ inf ku vkV þ j1 8 j10 C 2 ðMÞh ku uh kV þ j10 C 2 ðMÞh ku I h ukV ku uh kV v2V h
2
1 1 6 ð1 þ j9 Þ inf ku vkV þ j1 8 j10 C 2 ðMÞh ku uh kV þ j8 j10 C 2 ðMÞhkukH 4 ku uh kV : v2V h
Then for sufficiently small h, we have 2
1 ku uh kV 6 ð1 þ j9 Þ inf ku vkV þ j1 8 j10 C 2 ðMÞh ku uh kV : v2V h
Let 0 < h < 1. If 1 1 j1 8 j10 C 2 ðMÞh ku uh kV P h;
which also is 1
ku uh kV 6 ð1 hÞj8 j1 10 C 2 ðMÞ h; then ku uh kV 6 ð1 þ j9 Þh1 inf ku vkV 6 ch2 kukH 4 : v2V h
Summing up above, we obtain the H2-error estimate. Theorem 6. Suppose u and uh the solutions of (2) and (11) satisfying kukW 2;1 þ kuh kW 2;1 6 M: For sufficiently small h and u 2 W 4;2þe ðXÞ, where e is some positive constant, if u 2 DðR0 Þ and (10) holds, then we have ku uh kV 6 ch2 kukH 4 ; where c > 0 is independent of h. Remark 3. In Theorem 6, we require u 2 W 4;2þe ðXÞ for some e > 0. From Sobolev Embedding Theorem, then u 2 W 3;1 ðXÞ, which has been used in (19). If the solutions u and uh are sufficiently small, then from (17) and (19), we immediately obtain the following theorem. Theorem 7. Suppose u and uh the solutions of (2) and (11) satisfying e: kukW 2;1 þ kuh kW 2;1 6 M e is sufficiently small such that Suppose that u 2 DðR0 Þ. If M e ÞM e 6h j1 C 2 ð M 8
holds, then we have ku uh kV 6 ch2 kukH 4 ; where 0 < h < 1, c > 0 is independent of h. Proof. From (17) and (19), we have e e ku uh kV 6 ð1 þ j9 Þ inf ku vkV þ j1 8 C 2 ð M Þ M ku uh kV 6 ð1 þ j9 Þ inf ku vkV þ hku uh kV : v2V h
v2V h
Next, we prove L2-error estimate via duality technique. Consider the following biharmonic problem: 2 D w ¼ gðxÞ in X w ¼ ow ¼ 0 on oX: on
ð22Þ
R. An et al. / Applied Mathematics and Computation 194 (2007) 143–155
153
It is well known that kwkH 4 þ kwkV 6 ckgkL2 ; where c depends only on X. Setting w ¼ u uh , we have Z Z 2 ku uh kL2 ¼ ðu uh ; D2 wÞ ¼ Dðu uh ÞDðw wh Þdx þ Dðu uh ÞDwh dx X X Z Z 1 1 6 ku uh kV kw wh kV þ ð/ðjrujÞ /ðjruh jÞÞ/h dx ay rðu uh ÞDwh dx a X a X Z p 1 p 1 6 ku uh kV kw wh kV þ c ðjruj 2 þ jruh j 2 Þjru ruh k/h jdx X Z þ c jrðu uh ÞkDwh jdx X p 1
p 1
6 ch4 ku uh kL2 kukH 4 þ cðkukW2 1;1 þ kuh kW2 1;1 Þku uh kV kwh kL2 þ ckrðu uh ÞkL2 kwh kV 6 ch4 ku uh kL2 kukH 4 þ cM p2 1 h2 ku uh kV kwh kV þ krðu I h uÞkL2 kwh kV þ krðI h u uh ÞkL2 kwh kV 6 ch4 ku uh kL2 kukH 4 þ ch3 ku uh kL2 kukH 4 þ chku uh kL2 kuh I h ukV 6 ch3 ku uh kL2 kukH 4 þ chku uh kL2 kuh ukV þ chku uh kL2 ku I h ukV 6 ch3 ku uh kL2 kukH 4 : Summing up above, we obtain the L2-error estimate. Theorem 8. Under the assumptions of Theorem 6, for sufficiently small h, following L2-error estimate holds: ku uh kL2 6 ch3 kukH 4 ;
ð23Þ
where c > 0 is independent of h. Remark 4. The L2-error estimate (22) is not optimal. However, from the numerical experiments in Section 6, the convergence rate of L2-error is Oðh4 Þ if h is sufficiently small. In addition, if ay 0, then (22) can be improved to be optimal. Corollary 2. Under the assumptions of Theorem 8. If ay 0, then we have ku uh kL2 6 ch4 kukH 4 ; where c > 0 is independent of h. 6. Numerical results The numerical experiments in this section are designed to verity the error estimate obtained in Section 5. Here, we use Bogner–Fox–Schmidt element (Q3 Element). Consider the following fourth-order nonlinear problem: ( 2 4 DðDu þ 2jruj Þ ¼ jruj þ f ; in X; ð24Þ u ¼ ou ¼0 on oX; on where X ¼ fðx; yÞ; 0 6 x 6 1; 0 6 y 6 1g. Choose a true solution 2
2
u ¼ sinðpxÞ sinðpyÞ : It is easily to verify the u satisfying the homogeneous Dirichlet boundary conditions. Select u0 2 V satisfying ðDu0 ; DvÞ ¼ ðf ; vÞ
8v 2 V
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R. An et al. / Applied Mathematics and Computation 194 (2007) 143–155
as the initial value for iteration procedures. If un is known, then we compute unþ1 by the following simple iteration: ðDunþ1 ; DvÞ ¼ ðjrun j4 þ f ; vÞ ð2jrun j2 ; DvÞ
8v 2 V :
Denote M and N be the division numbers at x-axis and y-axis, respectively. Let K be the iteration numbers. Tables 1 and 2 display that the convergence rate of H2-error is almost Oðh2 Þ, which is consistent with Theorem 6. From Tables 3 and 4, we conclude that the optimal convergence order of L2-error on the problem (24) is obtained only for sufficiently small h. Table 1 The H2 relative error when M = N = H and K = 25 H
kuKh uK1 kV h kuK1 kV h
kuuh kV kukV
Order
8 16 32 64
0.997373244756572 0.998345931123661 0.998873146002898 0.999334728029242
7.967491672694477E002 1.994969017256056E002 4.997594771356409E003 1.313073616149285E003
n 2.00 2.00 1.93
Table 2 The H2 relative error when M = N = H and K = 30 H
kuKh uK1 kV h kuK1 kV h
kuuh kV kukV
Order
8 16 32 64
0.997372584913545 0.998344880491724 0.998874169146644 0.999336347862812
7.964694564277870E002 1.994549096190196E002 4.982081713114712E003 1.251366164489329E003
n 2.00 2.00 1.99
Table 3 The L2 relative error when M = N = H and K = 50 and ay 6¼ 0 H
kuKh uK1 kL2 h kuK1 kL2 h
kuuh kL2 kukL2
Order
8 16 32 64
0.990658875734683 0.977849077497979 0.930234820307248 0.807448430951118
8.773064381523671E003 8.832622608842352E004 5.230972374381978E005 3.245313951806216E006
n 3.31 4.08 4.01
Table 4 The L2 relative error when M = N = H and K = 60 and ay 6¼ 0 H
kuKh uK1 kL2 h kuK1 kL2 h
kuuh kL2 kukL2
Order
8 16 32 64
0.990658875111902 0.977849092199866 0.930234868567753 0.807448512100228
8.773066096053444E003 8.835136311855687E004 5.245861350625196E005 3.207989157622057E006
n 3.31 4.07 4.03
Table 5 The L2 relative error when M = N = H and K = 12 and ay 0 H
kuKh uK1 kL2 h kuK1 kL2 h
kuuh kL2 kukL2
Order
8 16 32 64
0.990624878485068 0.977799305369950 0.930212379795423 0.807440047682768
1.928800836470502E002 9.519844978571929E004 6.063129261860954E005 4.400933964318014E006
n 4.34 3.97 3.78
R. An et al. / Applied Mathematics and Computation 194 (2007) 143–155
155
Table 6 The L2 relative error when M = N = H and K = 15 and ay 0 H
kuKh uK1 kL2 h kuK1 kL2 h
kuuh kL2 kukL2
Order
8 16 32 64
0.990624878340708 0.977799305825940 0.930212385446254 0.807440138810543
1.928734477005187E002 9.513391037479355E004 5.998463989670770E005 3.737223748694858E006
n 4.34 3.99 4.00
Finally, we give a numerical experiment when ay 0. Consider ( 4 D2 u ¼ jruj þ f ; in X; u ¼ ou ¼0 on oX: on
ð25Þ
Tables 5 and 6 tell us that if ay 0, the optimal convergence order of L2-error is obtained. Acknowledgements Supported by the Nature Science Foundation of China (Grant Nos. 50306019, 10571142, 10471110 and 10471109). References [1] O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, 1969. [2] L.A. Peletier, W.C. Troy, Spatial patterns. Higher order models in physics and mechanics, Progress in Nonlinear Differerential Equations Applications, vol. 45, Birkhauser, Boston, MA, 2001. [3] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959) 623–727. [4] E. Berchio, F. Gazzola, Some remarks on biharmonic elliptic problems with positive, increasing and convex nonlinearities, Electron. J. Differ. Equat. 2005 (34) (2005) 1–20. [5] G. Arioli, F. Gazzola, H.C. Grunau, E. Mitidieri, A semilinear fourth order elliptic problem with exponential nonlinearity, SIAM J. Math. Anal. 36 (4) (2005) 1226–1258. [6] G. Arioli, F. Gazzola, H.C. Grunau, Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity, J. Differ. Equat. 230 (2006) 743–770. [7] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1998. [8] L.C. Evans, Partial Differential Equations, Providence American Mathematical Society, 1998. [9] Kaitai Li, Jian Su, Limin Gao, Optimal shape design for blade’s surface of an impeller via the Navier–Stokes equations, Commun. Numer. Meth. Eng. 22 (6) (2006) 657–676. [10] F. Brezzi, P. Raviart, Mixed finite element methods for fourth order elliptic equations, in: J. Miller (Ed.), Topic in Numerical Analysis III, Academic Press, New York, 1978. [11] P. Ciarlet, P. Raviart, A mixed finite element method for the biharmonic equation, in: C. de Boor (Ed.), Symposium on Mathematical Aspects of Finite Element in Partial Differential Equations, Academic Press, New York, 1974, pp. 125–143. [12] L. Herrmann, Finite element bending analysis for plates, J. Eng. Mech., Div. ASCE EM5 93 (1967) 49–83. [13] T. Miyoshi, A finite element method for the solution of fourth order partial differential equations, Kunamoto J. Sci. (Math.) 9 (1973) 87–116. [14] C. Johnson, On the convergence of a mixed finite element method for plate bending problems, Numer. Math. 21 (1973) 43–62. [15] I. Babuska, J. Osborn, J. Pitaranta, Analysis of mixed methods using mesh dependent norms, Math. Com. 35 (152) (1980) 1039–1062. [16] M.M. Karchevskii, A.D. Lyashko, M.R. Timerbaev, A mixed finite-element method for quasilinear degenerate fourth-order elliptic equations, Differential Equations 36 (7) (2000) 1050–1057. [17] M.M. Karchevskii, A.D. Lyashko, M.R. Timerbaev, A finite element method for fourth-order quasilinear degenerating elliptic equations, Differential Equations 35 (2) (1999) 233–238. [18] Zhangxing Chen, Finite Element Methods and Their Application, Spring-Verlag, 2005. [19] Jinchao Xu, Two-grid discretization techniques for linear and nonlinear PDE, SIAM J. Numer. Anal. 33 (1996) 1759–1777. [20] R. Rannacher, R. Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comput. 38 (1982) 437–445.