Journal of Computational and Applied Mathematics 7 () 7 3 www.elsevier.com/locate/cam A mixed nite volume element method based on rectangular mesh for biharmonic equations Tongke Wang College of Mathematical Science, Tianjin Normal University, Tianjin 37, PR China Received 3 January 3; received in revised form 5 February Abstract This paper presents a mixed nite volume element scheme based on rectangular partition for solving biharmonic equations. It also gives a kind of adaptive Uzawa iteration method for the scheme. It is rigorously proved that the scheme has rst-order accuracy in H semi-norm and L norm according to the characteristics of the scheme. Finally, two numerical examples illustrate the eectiveness of the method. c Elsevier B.V. All rights reserved. MSC: 65N Keywords: Biharmonic equation; Rectangular mesh; Mixed nite volume element method; Error estimate. Introduction Finite volume element method (FVEM) [,,] or its generalized form, nite volume method [5,9], uses a volume integral formulation of the dierential equation with a nite partitioning set of volume to discretize the equation. As far as the method is concerned, it is identical to the special case of the generalized dierence method (GDM) proposed by Professor Ronghua Li [6,8,,3,6], that is, linear or bilinear nite element space is used as trial or admissible nite element space and piecewise constant space is used as test function space. As for theoretical analysis, there are some dierences in FVEM and GDM. For example, FVEM conventionally estimates the error by discrete energy norm, whereas, GDM absorbs more ideas from nite element method. Because these methods keep conservation law of mass or energy, they are widely used in computational uid mechanics. In this paper, we are concerned about biharmonic equations. Because of their importance, lot of methods have been developed to treat biharmonic equations, for instance, 3-point nite dierence Corresponding author. Tel.: +86 33 33; fax: +86 35 6. E-mail address: wangtke@sina.com (T. Wang). 377-7/$ - see front matter c Elsevier B.V. All rights reserved. doi:.6/j.cam...
8 T. Wang / Journal of Computational and Applied Mathematics 7 () 7 3 scheme, higher-order nite element methods and more commonly used, mixed nite element methods [,3,7,,5]. As for generalized dierence methods, Wei Wu [] presented a kind of Ciarlet Raviart mixed generalized dierence method by triangulation and circumcenter dual partition. Zhongying Chen [6] proposed another method by other variational principle. Still based on Ciarlet Raviart mixed variational principle, we present a kind of mixed FVEM by rectangular partition in Section. In Section 3, rst-order error estimate is derived strictly in accordance with the characteristics of the FVEM. We note that the method of error estimate in this paper is somewhat dierent from the method in [8,]. We do not introduce the so-called Neumann projection and the method in this paper is more concise. Because the linear system of algebraic equations derived by mixed nite volume element scheme is indenite, classical iteration methods such as Gauss Seidel and SOR are not valid for the scheme. In Section, we construct a class of adaptive Uzawa iteration method. To verify the method in this paper, we compute some typical numerical examples and the results are very satisfactory. Compared with 3-point nite dierence scheme, the method in this paper has higher computation accuracy and can be used to solve more general problems.. Mixed FVEM Consider the following two-dimensional biharmonic equation on domain D: = f(x; y); (x; y) D; (.a) = 9 =; (x; y) 9D; 9n (.b) where f(x; y) is suciently smooth and n denotes the unit outward normal vector of 9D. For convenience, assume D =[; ]. By introducing vorticity =,(.a) is equivalent to = f; = : (.) Denote H m (D) by the standard Sobolevspace of order m. Also denote by H (m) (D) =H (D) H m (D). Let V D be any control volume with piecewise smooth boundary 9V. Integrate (.) over control volume V, then by Green s formula, the conservative integral form of (.) reads, nding ( ; ) H () (D) H (D), such that 9V 9V \9D 9 9n ds = V 9 9n ds = f dx dy V D; (.3a) V dx dy V D: (.3b) It is easy to prove that (.3) is equivalent to (.) for ( ; ) (C (D) C ( D)) C (D) and f C(D). In fact, from (.3a) and by Green s formula, we have ( f)dxdy = V D. From the V continuity of f and the arbitrariness of V D, we can derive = f in D. Restricting
T. Wang / Journal of Computational and Applied Mathematics 7 () 7 3 9 V D for (.3b), we know = in D, then 9V 9D (9 =9n)ds = for arbitrary V D. Ina similar argument, we know 9 =9n 9D =. Hence, (.) holds. The FVE approach of (.3) consists of replacing H () (D)(H (D)) by nite-dimensional space of piecewise smooth functions and using a nite set of volumes. In this paper, we consider rectangular partition of D and piecewise bilinear interpolations for and. First, give a nonuniform rectangular partition Q h for D and the nodes are (x i ;y j )(i=; ;:::;N;j= ; ;:::;M). Let D h ={(x i ;y j ); 6 i 6 N; 6 j 6 M} and denote all the interior nodes by D h. Further let h i = x i x i (i =; ;:::;N); k j = y j y j (j =; ;:::;M); x i = = x i h i =; x i+= = x i + h i+ =; y j = = y j k j =; y j+= = y j + k j+ =, then V ij =[x i = ;x i+= ] [y j = ;y j+= ] is a control volume or dual element of node (x i ;y j ). For boundary nodes, their control volumes should be modied correspondingly. For instance, V =[x ;x = ] [y ;y = ];V i =[x i = ;x i+= ] [y ;y = ] for i =; ;:::;N. All the control volumes constitute the dual partition Qh of domain D. Second, let H h H (D) and H h H (D) be both the piecewise bilinear nite element subspaces over partition Q h, then the mixed nite volume element scheme of (.3) reads, nding ( h ; h ) H h H h, such that 9V ij 9V ij\9d 9n ds = V ij f dx dy; i(j)=; ;:::;N (M ); (.a) 9n ds = h dx dy; i(j)=; ;:::;N(M): (.b) V ij Eq. (.) can be further written as dierence equations. Denote by ij = h (x i ;y j ); ij = h (x i ;y j ). For a uniform partition with M = N and h i = k j = h, (.) can be written as [ ij ( i ;j + i+;j + i+;j+ + i ;j+ ) ( i;j + i+;j + i;j+ + i ;j )] = f dx dy; i; j =; ;:::;N : (.5) V ij [ ij ( i ;j + i+;j + i+;j+ + i ;j+ ) ( i;j + i+;j + i;j+ + i ;j )] = h 6 [36 ij +( i ;j + i+;j + i+;j+ + i ;j+ ) +6( i;j + i+;j + i;j+ + i ;j )]; i;j =; ;:::;N : (.6) Bottom boundary (y = ): h 6 [9 +3 +3 + ]=3 ( + + ); h 6 [8 i +3 i ; +3 i+; + i ; +6 i + i+; ]
T. Wang / Journal of Computational and Applied Mathematics 7 () 7 3 = i ; +6 i; i+; i ; i; i+; ; i=; ;:::;N ; h 6 [9 N +3 N ; +3 N + N ; ]= N ; +3 N N ; N : (.7) Top boundary (y = ): h 6 [9 N +3 ;N +3 N + ;N ]=3 N ( ;N + N + ;N ); h 6 [8 in +3 i ;N +3 i+;n + i ;N +6 i;n + i+;n ] = i ;N +6 in i+;n i ;N i;n i+;n ; i=; ;:::;N ; h 6 [9 NN +3 N ;N +3 N;N + N ;N ] = N ;N +3 NN N ;N N;N : (.8) Left boundary (x = ): h 6 [8 j +3 ;j +3 ;j+ + ;j +6 j + ;j+ ] = ;j +6 ;j ;j+ ;j ;j ;j+ ; j =; ;:::;N : (.9) Right boundary (x = ): h 6 [8 Nj +3 N;j +3 N;j+ + N ;j +6 N ;j + N ;j+ ] = N;j +6 N;j N;j+ N ;j N ;j N ;j+ ; j =; ;:::;N : (.) Obviously, the linear system of equations (.5) (.) is simpler than that formed by mixed bilinear nite element method. It can be solved by Uzawa iteration method, see Section for detail. 3. Error estimate In Section, we derived a kind of nite volume element scheme. In this section, we further analyze the error of the scheme. Because the theory about the generalized dierence method [8,] has been established completely, we embed the scheme in Section into the theoretical framework of GDM. Suppose P(x i ;y j ) is an arbitrary node in D h. Denote V P = V ij by the corresponding dual element of node P and P by characteristic function over V P. Let h h = P Dh h (P) P h H h ; (3.)
T. Wang / Journal of Computational and Applied Mathematics 7 () 7 3 a(!; h h )= h (P) 9V P Dh P\9D 9! 9n ds;! H (D)(H () (D)); h H h (H h ); (!; h h )= h (P)! dx dy; h H h (H h ): (3.) V P P Dh By restricting the arbitrary control volume V to special V P, (.3) can be written as, nding ( ; ) H () (D) H (D), such that a(; h h )=(f; h h ) h H h ; (3.3a) a( ; h h )=(; h h ) h H h : (3.3b) Analogously, (.) equals to nding ( h ; h ) H h H h, such that a( h ; h h )=(f; h h ) h H h ; (3.a) a( h ; h h )=( h ; h h ) h H h : (3.b) Remark. For boundary node P and h H h, h (P) =. Hence, (3.3a) and (3.a) hold essentially for interior nodes and they just equal to (.3a) (V = V P ) and (.a), respectively. Suppose Q h is a quasi-uniformly regular partition, i.e., there exist constants ; ; 3 ;, satisfying max h i 6 min h i ; max k j 6 min k j ; i i j j 3 k j 6 h i 6 k j : (3.5) Let h = max(max h i ; max k j ). Depicted as in Fig., we convert the integral on the edge of dual partition to the related elements, then a( h ;h h )= [ h (P l ) h (P l+ )] E Q h l= M l Q 9 h 9n ds h H h ; h H h ; (3.6a) a( h ;h h )= [ h (P l ) h (P l+ )] E Q h l= M l Q 9n ds h H h ; h H h ; (3.6b) where P 5 = P. Remark. It is easy to see that (3.6a) also holds for h H (D) and (3.6b) holds for h H () (D). Denote s and s by continuous norm and continuous semi-norm of order s in Sobolevspace, respectively. Dene discrete H semi-norm and discrete L norm, respectively, by = = h ;h = h ;h;e ; h ;h = h ;h;e h H h (H h ); (3.7) E Q h E Q h
T. Wang / Journal of Computational and Applied Mathematics 7 () 7 3 Fig.. Illustration for an element and its nodes. where E = P P P 3 P =[x i ;x i ] [y j ;y j ], shown as in Fig. and h ;h;e = k j ( h (P l+ ) h (P l )) + h i ( h (P l+ ) h (P l )) ; h i k j h ;h;e = h ik j l=;3 h (P l ) : l= l=; Lemma. For h H h (H h ); h ;h is equivalent to h and h ;h is equivalent to h, that is, the following inequalities hold: 3 3 h ;h 6 h 6 h ;h ; 3 h ;h 6 h 6 h ;h : (3.8) Proof. Suppose Q is the center of element E. Let =(x x Q )=h i ; =(y y Q )=k j, then E is transformed to Ê =[ ; ]. Construct bilinear interpolating base functions on Ê, which are N = ( )( ); N = ( + )( ); N 3 = ( + )( + ); N = ( )( + ): Then h = l= h(p l )N l (; ). According to the denition of h ;E, we have [ ( ) h k j 9 h ;E = + h ( ) ] i 9 h d d h i 9 k j 9 Ê = k j 3h i [( h (P ) h (P )) +( h (P 3 ) h (P )) +( h (P ) h (P ))( h (P 3 ) h (P ))] + h i 3k j [( h (P ) h (P )) +( h (P 3 ) h (P )) +( h (P ) h (P ))( h (P 3 ) h (P ))]:
T. Wang / Journal of Computational and Applied Mathematics 7 () 7 3 3 By Cauchy inequality and (3.7), the rst inequality of (3.8) is proved. As for the second one, a straightforward computing shows h (P ) h ;E = h ik j 36 [ h(p ); h (P ); h (P 3 ); h (P )] h (P ) h (P 3 ) : h (P ) The eigenvalues of the matrix of the right-hand side of the above formula are l =; 3; 3; 9, from which we can obtain 9 h ;h 6 h 6 h ;h: Lemma is proved. Lemma. a( h ; h h )=a( h ; h h ) h H h h H h ; (3.9) a( h ; h h ) h ;h h h H h : (3.) Proof. By (3.6), further computing the integrals, we have M Q 9n ds = k j [3( h (P ) h (P ))+( h (P 3 ) h (P ))]; 8h i M 3Q 9n ds = k j [( h (P ) h (P ))+3( h (P 3 ) h (P ))]; 8h i M Q 9n ds = h i [( h (P ) h (P ))+3( h (P 3 ) h (P ))]; 8k j M Q 9n ds = h i [3( h (P ) h (P ))+( h (P 3 ) h (P ))]: 8k j The integrals about h can be obtained analogously. A straightforward verication shows 9 h [ h (P l ) h (P l+ )] l=;3 M l Q 9n ds = [ h (P l ) h (P l+ )] l=;3 M l Q 9n ds: Similar formula can also be obtained for l =;. Thus, (3.9) holds. As for (3.), we have { } a( h ;h h ) k j ( h (P l+ ) h (P l )) + h i ( h (P l+ ) h (P l )) h i k j The proof is completed. E Q h l=;3 = h ;h h : l=;
T. Wang / Journal of Computational and Applied Mathematics 7 () 7 3 Lemma 3. ( h ; h h ) h ;h h h H h ; (3.a) (!; h h ) 6! h ;h 6 3! h! H (D) h H h : (3.b) Proof. ( h ;h h )= h (P) h dx dy = h (P l ) h dx dy; V P P Dh E Q h l= V Pl E where for example h dx dy = h ik j V P E 6 [9 h(p )+3 h (P )+ h (P 3 )+3 h (P )]: Analogously, we can get the other integrals. Add these integrals, then h (P ) h (P l ) h dx dy = h ik j l= V Pl E 6 [ h(p ); h (P ); h (P 3 ); h (P )]M h (P ) h (P 3 ) ; h (P ) where 9 3 3 3 9 3 M = 3 9 3 : 3 3 9 A simple computation shows the eigenvalues of matrix M are l =; 8; 8; 6, from which we can get (3.a). As for (3.b), denote S VP by the area of the dual element V P, then (!; h h ) 6 = h (P) S VP = ( )! dx dy 6! h ;h : S VP V P Dh P P Dh By (3.8) we know (3.b) holds. Lemma 3 is proved. From Lemmas and 3, we know that scheme (3.) has a unique solution. In fact, without loss of generality, assume f in(3.a). Let h = h in (3.a) and h = h in (3.b), then by (3.9), we have ( h ; h h)=.by(3.a), we can derive h. In particular, taking h = h in (3.b) and by (3.), we know h. That is, the solution of (3.) is unique. Let h :H (D) H h and h :H (D) H h be two piecewise bilinear interpolation projects. By the interpolation theory in Sobolevspace, we have h 6 Ch ; h 6 Ch : (3.)
T. Wang / Journal of Computational and Applied Mathematics 7 () 7 3 5 Lemma. Assume H (D) H 3 (D) and H (D), then there exists a positive constant C, independent of the mesh size h, such that a( h ; h h ) 6 Ch 3 h ; a( h ; h h ) 6 Ch h h H h ; (3.3) a( h ; h h ) 6 Ch h h H h : (3.) Proof. From (3.6b) and Remark, noting Fig., we have a( h ; h h )= [ h (P l ) h (P l+ )] E Q h M Q 9( h ) 9n ds = k j h i l= { 9 (;) 9 M l Q 9( h ) 9n ds; (3.5) d } 8 [3( (P ) (P ))+( (P 3 ) (P ))], k j I ( ): h i As a linear functional of H 3 (D); I ( ) satises I ( ) 6 C. In addition, H 3 ; ;Ê (Ê), C (Ê). Hence, I ( ) 6 C. A straightforward calculation shows I 3;Ê ( ) for =;;; ; ;. By Bramble Hilbert Lemma [], we know I ( ) 6 C 3;Ê. By an integral transformation, we have I ( ) 6 Ch 3;E. Thus, 9( h ) ds M Q 9n 6 C k j h 3;E : h i The other integrals in (3.5) have similar estimates. Using the inequality ( ) ( = ) = ai b i 6 i a i b i ; i noting (3.5), we get [ h (P l ) h (P l+ )] l= M l Q 9( h ) 9n ds 6 Ch 3;E h ;h;e : Again from (3.5), by Cauchy inequality and Lemma, we obtain the rst inequality of (3.3). Now we prove the second one of (3.3). I ( ) is still dened as above. Using trace theorem, we further have I ( ) 6 C( ;Ê + ; ;Ê ): Because H (Ê), C (Ê), thus, I ( ) 6 C ;Ê. Again by Bramble Hilbert Lemma, we can prove the second inequality of (3.3). As for (3.), from (3.6a), a formula analogous to (3.5) can be derived. Thus, (3.) holds. Lemma is proved. Lemma 5 (Ciarlet []). For h H h ; h H h, we have h 6 Ch h ; h 6 C h : (3.6)
6 T. Wang / Journal of Computational and Applied Mathematics 7 () 7 3 Subtracting (3.) from (3.3), we obtain the following error equations: a( h ; h h )= h H h ; (3.7a) a( h ; h h )=( h ; h h ) h H h : (3.7b) Now we state the main result of this section. Theorem. Assume ( ; ) (H (D) H 3 (D)) H (D) is the solution of (.3) and Q h is a quasiuniformly rectangular partition of domain D, then the approximate solution ( h ; h ) of mixed nite volume element scheme (.) converges to the true solution ( ; ) with the following estimate: h + h 6 Ch( 3 + + + h ): (3.8) Proof. From Lemmas to 5 and by -Cauchy inequality, we have Hence, h h 6 ( h h ; h( h h )) =[a( h ; h( h h )) + a( h h ; h( h h )) ( h ; h( h h ))] =[a( h ; h( h h )) a( h ; h( h h )) ( h ; h( h h ))] 6 Ch 3 h h + Ch h h + Ch h h 6 Ch 3 h h + Ch h h + Ch h h 6 Ch 3 + Ch + Ch + C h h + h h : h h 6 Ch 3 + Ch + Ch + C h h : (3.9) Because H h H h,(3.7b) also holds for H h, thus h h 6 a( h h ; h( h h )) =[( h ; h( h h ))+( h h ; h( h h )) a( h ; h( h h ))] 6 Ch h h + C h h h h + Ch h h : By Lemma 5, we obtain h h 6 C h h + Ch + Ch : (3.)
T. Wang / Journal of Computational and Applied Mathematics 7 () 7 3 7 Substituting (3.) into (3.9) and taking C =, we have h h 6 Ch( 3 + + + h ): (3.) Further substitute (3.) into (3.), then h h 6 Ch( 3 + + + h ): (3.) Error estimate (3.8) now follows directly from interpolating error estimate (3.). The proof is complete.. Adaptive Uzawa iteration algorithm and numerical experiment Because the linear system of equations derived by (.) is a typical indenite one, we must adopt the methods which are suitable for this problem. Here we use Uzawa iteration method [] to solve (.). Denote N by the number of boundary nodes of Q h and i (i=; ;:::;N ), the piecewise bilinear interpolating base functions corresponding to the boundary nodes. Let M h =span{ ; ;:::; N }, then H h = H h M h. For ; M h, dene their inner product by N N N (; ) Mh = k k k dx dy; where = k k ; = k k : (.) V k k= Then the adaptive Uzawa iteration algorithm can be stated as follows: k=. Given arbitrary h M h and. Let norm =.. Assume h l M h is known, nd h l, such that h l h l H h ; a(h; l h h )=(f; h h ) h H h : (.) 3. Find h l H h, such that a( h; l h h )=(h; l h h ) h H h : (.3). Solve l+ h, which satises ( l+ h h; l h ) Mh = [a( h; l h ) (h; l h )] h M h : (.) 5. Compute norm =( l+ h h l;l+ h h l)=. If norm norm, then set =. 6. If norm 6 7, stop; otherwise, norm norm, repeat 6. From the theory of Uzawa iteration method [,], there exists max such that the above iteration procedure is convergent. It is usually dicult to compute max, so, we write the above procedure as adaptive one. In the following we provide two numerical examples to illustrate the eectiveness of scheme (.). k= Example. Let D =[;] and f(x; y) = 6 cos(x) cos(y) cos(x) cos(y) in(.a), then the true solution to (.) is =(sin x) (sin y). Let h==3, compute this problem by mixed FVEM (.) and the results are shown in Fig.. We also compute the maximum absolute errors of and
8 T. Wang / Journal of Computational and Applied Mathematics 7 () 7 3.8 3 Psi.6. Omega. - 3 y x 3-3 y x 3 Fig.. The results computed by scheme (.) in Example : left: the computational surface of ; right: the computational surface of., respectively, which are E =:77 3 and E =:83, where E = h and E = h. For comparison, we further compute this problem by 3-point nite dierence scheme (denote by FDS3P) and the results are E =6:8787 3 and E =:785, from which we know the accuracy of the scheme in this paper is obviously higher than that of 3-point nite dierence scheme. Example. Compute the deections of the thin clamped unit square plate []. Consider two cases: case A and case B. In case A, the load is uniform and we take f(x; y) =. From [], we know ( ; ) :65 3. In case B, we choose f = (x ;y ), where being the Delta function. Thus, in case B, the plate is under the action of a concentrated central load. Also from [], ( ; ) 5:6 3. In case A, we choose h =:5 and.5 and compute the problem by scheme (.) and 3-point nite dierence scheme (FDS3P). In case B, because (x ;y ) tends innite at ( ; ), it is impossible to use FDS3P. Only scheme (.) is implemented. For cases A and B, the approximate solution of ( ; ) is shown in Table. We further plot the approximate solutions of and, depicted as in Figs. 3 and for cases A and B, respectively. From Examples and, we know that the scheme in this paper gets very satisfactory results for dierent load f(x; y) and the scheme can be well applied to solve biharmonic equations. Table The approximate solution of ( ; ) computed by some schemes in Example h =:5 h =:5 Case A Case B Case A Case B MFVEM (.) :79 3 5:7 3 :685 3 5:6 3 FDS3P :979 3 :39 3
T. Wang / Journal of Computational and Applied Mathematics 7 () 7 3 9 x 3.5.. Psi.5 Omega...5 y.5 x.6.5 y.5 x Fig. 3. The results computed by scheme (.) in Example (case A, h = :5): left: the computational surface of ; right: the computational surface of. x 3 6.8 5.6 Psi 3 Omega...5 y.5 x..5 y.5 x Fig.. The results computed by scheme (.) in Example (case B, h = :5): left: the computational surface of ; right: the computational surface of. Acknowledgements The author is very grateful to Professor Yuan Yirang of Shandong University for his encouragement and to referees for their valuable remarks, which helped improve the presentation substantially. References [] P.G. Ciarlet, The Finite Element Methods for Elliptic Problems, North-Holland, Amsterdam, 978. [] P.G. Ciarlet, P. Raviart, A mixed nite element method for the biharmonic equations, in: C. de Boor (Ed.), Symposium on Mathematical Aspects of Finite Elements in Partial Dierential Equations, Academic Press, New York, 97, pp. 5 5. [3] R.S. Falk, J.E. Osborn, Error estimates for mixed methods, RAIRO, Anal. Numer. (98) 9 77. [] Jianguo Huang, Shitong Xi, On the nite volume elements method for general self-adjoint elliptic problems, SIAM J. Numer. Anal. 35 (5) (998) 76 77.
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