A mixed nite volume element method based on rectangular mesh for biharmonic equations
|
|
- Fay Pope
- 5 years ago
- Views:
Transcription
1 Journal of Computational and Applied Mathematics 7 () 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.: ; fax: address: wangtke@sina.com (T. Wang) /$ - see front matter c Elsevier B.V. All rights reserved. doi:.6/j.cam...
2 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
3 T. Wang / Journal of Computational and Applied Mathematics 7 () 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 [ ]=3 ( + + ); h 6 [8 i +3 i ; +3 i+; + i ; +6 i + i+; ]
4 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.)
5 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
6 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 ))]:
7 T. Wang / Journal of Computational and Applied Mathematics 7 () 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=;
8 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 M = : 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.)
9 T. Wang / Journal of Computational and Applied Mathematics 7 () 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)
10 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( 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.)
11 T. Wang / Journal of Computational and Applied Mathematics 7 () Substituting (3.) into (3.9) and taking C =, we have h h 6 Ch( h ): (3.) Further substitute (3.) into (3.), then h h 6 Ch( 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
12 8 T. Wang / Journal of Computational and Applied Mathematics 7 () 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: 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 : :6 3 FDS3P :979 3 :39 3
13 T. Wang / Journal of Computational and Applied Mathematics 7 () x 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 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 [3] R.S. Falk, J.E. Osborn, Error estimates for mixed methods, RAIRO, Anal. Numer. (98) [] Jianguo Huang, Shitong Xi, On the nite volume elements method for general self-adjoint elliptic problems, SIAM J. Numer. Anal. 35 (5) (998)
14 3 T. Wang / Journal of Computational and Applied Mathematics 7 () 7 3 [5] W.P. Jones, K.R. Menziest, Analysis of the cell-centred nite volume method for the diusion equation, J. Comput. Phys. 65 () [6] Peiqi Zhu, Ronghua Li, Generalized dierence methods for nd order elliptic partial dierential equations (II), Numer. Math. J. Chinese Univ. () (98) [7] Qun Lin, Jichun Li, Aihui Zhou, A rectangle test for Ciarlet Raviart and Herrmann Miyoshi schemes, Proceedings of the Systems Science and Systems Engineering, Great Wall Culture Publ. Co., Hongkong, 99, pp [8] Ronghua Li, Zhongying Chen, Wei Wu, Generalized Dierence Methods for Dierential Equations Numerical Analysis of Finite Volume Methods, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 6, Marcel Dekker Inc., New York,. [9] V. Selmin, The node-centred nite volume approach: bridge between nite dierences and nite elements, Comput. Methods Appl. Mech. Engrg. (993) [] E. Suli, Convergence of nite volume schemes for Poisson s equation on nonuniform meshes, SIAM J. Numer. Anal. 8 (5) (99) 9 3. [] Wei Wu, The mixed generalized dierence method for biharmonic equations, J. Jilin Univ. (Nat. Sci.) (986). [] Xiao-liang Cheng, Weimin Han, Hong-ci Huang, Some mixed nite element methods for biharmonic equation, J. Comput. Appl. Math. 6 () 9 9. [3] Yonghai Li, Ronghua Li, Generalized dierence methods on arbitrary quadrilateral networks, J. Comput. Math. 7 (6) (999) [] Zhiqiang Cai, S. McCormick, On the accuracy of the nite volume element method for diusion equations on composite grid, SIAM J. Numer. Anal. 7 (3) (99) [5] Zhiyuan Pang, Error estimates for mixed nite element methods, Math. Numer. Sinica (986) [6] Zhongying Chen, Variational principle and numerical analysis of generalized dierence method for biharmonic equation, Numer. Math. J. Chinese Univ. 5 () (993) 8 9.
A MULTIGRID ALGORITHM FOR. Richard E. Ewing and Jian Shen. Institute for Scientic Computation. Texas A&M University. College Station, Texas SUMMARY
A MULTIGRID ALGORITHM FOR THE CELL-CENTERED FINITE DIFFERENCE SCHEME Richard E. Ewing and Jian Shen Institute for Scientic Computation Texas A&M University College Station, Texas SUMMARY In this article,
More informationAn explicit nite element method for convection-dominated compressible viscous Stokes system with inow boundary
Journal of Computational and Applied Mathematics 156 (2003) 319 343 www.elsevier.com/locate/cam An explicit nite element method for convection-dominated compressible viscous Stokes system with inow boundary
More informationMULTIGRID PRECONDITIONING FOR THE BIHARMONIC DIRICHLET PROBLEM M. R. HANISCH
MULTIGRID PRECONDITIONING FOR THE BIHARMONIC DIRICHLET PROBLEM M. R. HANISCH Abstract. A multigrid preconditioning scheme for solving the Ciarlet-Raviart mixed method equations for the biharmonic Dirichlet
More informationA Mixed Nonconforming Finite Element for Linear Elasticity
A Mixed Nonconforming Finite Element for Linear Elasticity Zhiqiang Cai, 1 Xiu Ye 2 1 Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395 2 Department of Mathematics and Statistics,
More informationOn an Approximation Result for Piecewise Polynomial Functions. O. Karakashian
BULLETIN OF THE GREE MATHEMATICAL SOCIETY Volume 57, 010 (1 7) On an Approximation Result for Piecewise Polynomial Functions O. arakashian Abstract We provide a new approach for proving approximation results
More informationon! 0, 1 and 2 In the Zienkiewicz-Zhu SPR p 1 and p 2 are obtained by solving the locally discrete least-squares p
Analysis of a Class of Superconvergence Patch Recovery Techniques for Linear and Bilinear Finite Elements Bo Li Zhimin Zhang y Abstract Mathematical proofs are presented for the derivative superconvergence
More informationLecture Note III: Least-Squares Method
Lecture Note III: Least-Squares Method Zhiqiang Cai October 4, 004 In this chapter, we shall present least-squares methods for second-order scalar partial differential equations, elastic equations of solids,
More informationIt is known that Morley element is not C 0 element and it is divergent for Poisson equation (see [6]). When Morley element is applied to solve problem
Modied Morley Element Method for a ourth Order Elliptic Singular Perturbation Problem Λ Wang Ming LMAM, School of Mathematical Science, Peking University Jinchao u School of Mathematical Science, Peking
More informationFinite element approximation on quadrilateral meshes
COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING Commun. Numer. Meth. Engng 2001; 17:805 812 (DOI: 10.1002/cnm.450) Finite element approximation on quadrilateral meshes Douglas N. Arnold 1;, Daniele
More informationNovel determination of dierential-equation solutions: universal approximation method
Journal of Computational and Applied Mathematics 146 (2002) 443 457 www.elsevier.com/locate/cam Novel determination of dierential-equation solutions: universal approximation method Thananchai Leephakpreeda
More informationquantitative information on the error caused by using the solution of the linear problem to describe the response of the elastic material on a corner
Quantitative Justication of Linearization in Nonlinear Hencky Material Problems 1 Weimin Han and Hong-ci Huang 3 Abstract. The classical linear elasticity theory is based on the assumption that the size
More informationPARTITION OF UNITY FOR THE STOKES PROBLEM ON NONMATCHING GRIDS
PARTITION OF UNITY FOR THE STOES PROBLEM ON NONMATCHING GRIDS CONSTANTIN BACUTA AND JINCHAO XU Abstract. We consider the Stokes Problem on a plane polygonal domain Ω R 2. We propose a finite element method
More informationHigh order, finite volume method, flux conservation, finite element method
FLUX-CONSERVING FINITE ELEMENT METHODS SHANGYOU ZHANG, ZHIMIN ZHANG, AND QINGSONG ZOU Abstract. We analyze the flux conservation property of the finite element method. It is shown that the finite element
More informationA Least-Squares Finite Element Approximation for the Compressible Stokes Equations
A Least-Squares Finite Element Approximation for the Compressible Stokes Equations Zhiqiang Cai, 1 Xiu Ye 1 Department of Mathematics, Purdue University, 1395 Mathematical Science Building, West Lafayette,
More informationNon-Conforming Finite Element Methods for Nonmatching Grids in Three Dimensions
Non-Conforming Finite Element Methods for Nonmatching Grids in Three Dimensions Wayne McGee and Padmanabhan Seshaiyer Texas Tech University, Mathematics and Statistics (padhu@math.ttu.edu) Summary. In
More informationComparison results between Jacobi and other iterative methods
Journal of Computational and Applied Mathematics 169 (2004) 45 51 www.elsevier.com/locate/cam Comparison results between Jacobi and other iterative methods Zhuan-De Wang, Ting-Zhu Huang Faculty of Applied
More informationETNA Kent State University
Electronic Transactions on Numerical Analysis. Volume 37, pp. 166-172, 2010. Copyright 2010,. ISSN 1068-9613. ETNA A GRADIENT RECOVERY OPERATOR BASED ON AN OBLIQUE PROJECTION BISHNU P. LAMICHHANE Abstract.
More informationA Finite Element Method for an Ill-Posed Problem. Martin-Luther-Universitat, Fachbereich Mathematik/Informatik,Postfach 8, D Halle, Abstract
A Finite Element Method for an Ill-Posed Problem W. Lucht Martin-Luther-Universitat, Fachbereich Mathematik/Informatik,Postfach 8, D-699 Halle, Germany Abstract For an ill-posed problem which has its origin
More informationA Finite Element Method Using Singular Functions for Poisson Equations: Mixed Boundary Conditions
A Finite Element Method Using Singular Functions for Poisson Equations: Mixed Boundary Conditions Zhiqiang Cai Seokchan Kim Sangdong Kim Sooryun Kong Abstract In [7], we proposed a new finite element method
More informationXIAO-CHUAN CAI AND MAKSYMILIAN DRYJA. strongly elliptic equations discretized by the nite element methods.
Contemporary Mathematics Volume 00, 0000 Domain Decomposition Methods for Monotone Nonlinear Elliptic Problems XIAO-CHUAN CAI AND MAKSYMILIAN DRYJA Abstract. In this paper, we study several overlapping
More informationMULTIGRID PRECONDITIONING IN H(div) ON NON-CONVEX POLYGONS* Dedicated to Professor Jim Douglas, Jr. on the occasion of his seventieth birthday.
MULTIGRID PRECONDITIONING IN H(div) ON NON-CONVEX POLYGONS* DOUGLAS N ARNOLD, RICHARD S FALK, and RAGNAR WINTHER Dedicated to Professor Jim Douglas, Jr on the occasion of his seventieth birthday Abstract
More informationTHE BEST L 2 NORM ERROR ESTIMATE OF LOWER ORDER FINITE ELEMENT METHODS FOR THE FOURTH ORDER PROBLEM *
Journal of Computational Mathematics Vol.30, No.5, 2012, 449 460. http://www.global-sci.org/jcm doi:10.4208/jcm.1203-m3855 THE BEST L 2 NORM ERROR ESTIMATE OF LOWER ORDER FINITE ELEMENT METHODS FOR THE
More information290 J.M. Carnicer, J.M. Pe~na basis (u 1 ; : : : ; u n ) consisting of minimally supported elements, yet also has a basis (v 1 ; : : : ; v n ) which f
Numer. Math. 67: 289{301 (1994) Numerische Mathematik c Springer-Verlag 1994 Electronic Edition Least supported bases and local linear independence J.M. Carnicer, J.M. Pe~na? Departamento de Matematica
More informationGeometric Multigrid Methods
Geometric Multigrid Methods Susanne C. Brenner Department of Mathematics and Center for Computation & Technology Louisiana State University IMA Tutorial: Fast Solution Techniques November 28, 2010 Ideas
More informationMathematics Research Report No. MRR 003{96, HIGH RESOLUTION POTENTIAL FLOW METHODS IN OIL EXPLORATION Stephen Roberts 1 and Stephan Matthai 2 3rd Febr
HIGH RESOLUTION POTENTIAL FLOW METHODS IN OIL EXPLORATION Stephen Roberts and Stephan Matthai Mathematics Research Report No. MRR 003{96, Mathematics Research Report No. MRR 003{96, HIGH RESOLUTION POTENTIAL
More informationMultigrid Methods for Saddle Point Problems
Multigrid Methods for Saddle Point Problems Susanne C. Brenner Department of Mathematics and Center for Computation & Technology Louisiana State University Advances in Mathematics of Finite Elements (In
More informationMIXED FINITE ELEMENT METHODS FOR PROBLEMS WITH ROBIN BOUNDARY CONDITIONS
MIXED FINITE ELEMENT METHODS FOR PROBLEMS WITH ROBIN BOUNDARY CONDITIONS JUHO KÖNNÖ, DOMINIK SCHÖTZAU, AND ROLF STENBERG Abstract. We derive new a-priori and a-posteriori error estimates for mixed nite
More informationSimple Examples on Rectangular Domains
84 Chapter 5 Simple Examples on Rectangular Domains In this chapter we consider simple elliptic boundary value problems in rectangular domains in R 2 or R 3 ; our prototype example is the Poisson equation
More informationUltraconvergence of ZZ Patch Recovery at Mesh Symmetry Points
Ultraconvergence of ZZ Patch Recovery at Mesh Symmetry Points Zhimin Zhang and Runchang Lin Department of Mathematics, Wayne State University Abstract. The ultraconvergence property of the Zienkiewicz-Zhu
More informationarxiv: v1 [math.na] 29 Feb 2016
EFFECTIVE IMPLEMENTATION OF THE WEAK GALERKIN FINITE ELEMENT METHODS FOR THE BIHARMONIC EQUATION LIN MU, JUNPING WANG, AND XIU YE Abstract. arxiv:1602.08817v1 [math.na] 29 Feb 2016 The weak Galerkin (WG)
More informationConstrained Leja points and the numerical solution of the constrained energy problem
Journal of Computational and Applied Mathematics 131 (2001) 427 444 www.elsevier.nl/locate/cam Constrained Leja points and the numerical solution of the constrained energy problem Dan I. Coroian, Peter
More informationFinite Elements. Colin Cotter. February 22, Colin Cotter FEM
Finite Elements February 22, 2019 In the previous sections, we introduced the concept of finite element spaces, which contain certain functions defined on a domain. Finite element spaces are examples of
More informationBasic Principles of Weak Galerkin Finite Element Methods for PDEs
Basic Principles of Weak Galerkin Finite Element Methods for PDEs Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 Polytopal Element
More informationHamburger Beiträge zur Angewandten Mathematik
Hamburger Beiträge zur Angewandten Mathematik Numerical analysis of a control and state constrained elliptic control problem with piecewise constant control approximations Klaus Deckelnick and Michael
More informationCrouzeix-Velte Decompositions and the Stokes Problem
Crouzeix-Velte Decompositions and te Stokes Problem PD Tesis Strauber Györgyi Eötvös Loránd University of Sciences, Insitute of Matematics, Matematical Doctoral Scool Director of te Doctoral Scool: Dr.
More informationA POSTERIORI ERROR ESTIMATES OF TRIANGULAR MIXED FINITE ELEMENT METHODS FOR QUADRATIC CONVECTION DIFFUSION OPTIMAL CONTROL PROBLEMS
A POSTERIORI ERROR ESTIMATES OF TRIANGULAR MIXED FINITE ELEMENT METHODS FOR QUADRATIC CONVECTION DIFFUSION OPTIMAL CONTROL PROBLEMS Z. LU Communicated by Gabriela Marinoschi In this paper, we discuss a
More informationSpectral gradient projection method for solving nonlinear monotone equations
Journal of Computational and Applied Mathematics 196 (2006) 478 484 www.elsevier.com/locate/cam Spectral gradient projection method for solving nonlinear monotone equations Li Zhang, Weijun Zhou Department
More informationAn anisotropic, superconvergent nonconforming plate finite element
Journal of Computational and Applied Mathematics 220 (2008 96 0 www.elsevier.com/locate/cam An anisotropic, superconvergent nonconforming plate finite element Shaochun Chen a,liyin a, Shipeng Mao b, a
More informationThe reflexive and anti-reflexive solutions of the matrix equation A H XB =C
Journal of Computational and Applied Mathematics 200 (2007) 749 760 www.elsevier.com/locate/cam The reflexive and anti-reflexive solutions of the matrix equation A H XB =C Xiang-yang Peng a,b,, Xi-yan
More informationNew Lyapunov Krasovskii functionals for stability of linear retarded and neutral type systems
Systems & Control Letters 43 (21 39 319 www.elsevier.com/locate/sysconle New Lyapunov Krasovskii functionals for stability of linear retarded and neutral type systems E. Fridman Department of Electrical
More information1. Introduction. We consider the model problem that seeks an unknown function u = u(x) satisfying
A SIMPLE FINITE ELEMENT METHOD FOR LINEAR HYPERBOLIC PROBLEMS LIN MU AND XIU YE Abstract. In this paper, we introduce a simple finite element method for solving first order hyperbolic equations with easy
More informationConstruction of `Wachspress type' rational basis functions over rectangles
Proc. Indian Acad. Sci. (Math. Sci.), Vol. 110, No. 1, February 2000, pp. 69±77. # Printed in India Construction of `Wachspress type' rational basis functions over rectangles P L POWAR and S S RANA Department
More informationNonmonotonic back-tracking trust region interior point algorithm for linear constrained optimization
Journal of Computational and Applied Mathematics 155 (2003) 285 305 www.elsevier.com/locate/cam Nonmonotonic bac-tracing trust region interior point algorithm for linear constrained optimization Detong
More informationOn the convergence of interpolatory-type quadrature rules for evaluating Cauchy integrals
Journal of Computational and Applied Mathematics 49 () 38 395 www.elsevier.com/locate/cam On the convergence of interpolatory-type quadrature rules for evaluating Cauchy integrals Philsu Kim a;, Beong
More informationImproved Newton s method with exact line searches to solve quadratic matrix equation
Journal of Computational and Applied Mathematics 222 (2008) 645 654 wwwelseviercom/locate/cam Improved Newton s method with exact line searches to solve quadratic matrix equation Jian-hui Long, Xi-yan
More informationwith Applications to Elasticity and Compressible Flow Daoqi Yang March 20, 1997 Abstract
Stabilized Schemes for Mixed Finite Element Methods with Applications to Elasticity and Compressible Flow Problems Daoqi Yang March 20, 1997 Abstract Stabilized iterative schemes for mixed nite element
More informationNodal O(h 4 )-superconvergence of piecewise trilinear FE approximations
Preprint, Institute of Mathematics, AS CR, Prague. 2007-12-12 INSTITTE of MATHEMATICS Academy of Sciences Czech Republic Nodal O(h 4 )-superconvergence of piecewise trilinear FE approximations Antti Hannukainen
More informationSUPERCONVERGENCE PROPERTIES FOR OPTIMAL CONTROL PROBLEMS DISCRETIZED BY PIECEWISE LINEAR AND DISCONTINUOUS FUNCTIONS
SUPERCONVERGENCE PROPERTIES FOR OPTIMAL CONTROL PROBLEMS DISCRETIZED BY PIECEWISE LINEAR AND DISCONTINUOUS FUNCTIONS A. RÖSCH AND R. SIMON Abstract. An optimal control problem for an elliptic equation
More informationA WEAK GALERKIN MIXED FINITE ELEMENT METHOD FOR BIHARMONIC EQUATIONS
A WEAK GALERKIN MIXED FINITE ELEMENT METHOD FOR BIHARMONIC EQUATIONS LIN MU, JUNPING WANG, YANQIU WANG, AND XIU YE Abstract. This article introduces and analyzes a weak Galerkin mixed finite element method
More informationTo link to this article:
This article was downloaded by: [Wuhan University] On: 11 March 2015, At: 01:08 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer
More informationAn interpolation operator for H 1 functions on general quadrilateral and hexahedral meshes with hanging nodes
An interpolation operator for H 1 functions on general quadrilateral and hexahedral meshes with hanging nodes Vincent Heuveline Friedhelm Schieweck Abstract We propose a Scott-Zhang type interpolation
More informationA new ane scaling interior point algorithm for nonlinear optimization subject to linear equality and inequality constraints
Journal of Computational and Applied Mathematics 161 (003) 1 5 www.elsevier.com/locate/cam A new ane scaling interior point algorithm for nonlinear optimization subject to linear equality and inequality
More informationAbstract. 1. Introduction
Journal of Computational Mathematics Vol.28, No.2, 2010, 273 288. http://www.global-sci.org/jcm doi:10.4208/jcm.2009.10-m2870 UNIFORM SUPERCONVERGENCE OF GALERKIN METHODS FOR SINGULARLY PERTURBED PROBLEMS
More informationNumerical Integration over Pyramids
Numerical Integration over Pyramids Chuan Miao Chen, Michal řížek, Liping Liu Corresponding Author: Dr. Liping Liu Department of Mathematical Sciences Lakehead University Thunder Bay, ON Canada P7B 5E1
More informationComparison of V-cycle Multigrid Method for Cell-centered Finite Difference on Triangular Meshes
Comparison of V-cycle Multigrid Method for Cell-centered Finite Difference on Triangular Meshes Do Y. Kwak, 1 JunS.Lee 1 Department of Mathematics, KAIST, Taejon 305-701, Korea Department of Mathematics,
More informationConvergence of A Galerkin Method for 2-D Discontinuous Euler Flows Jian-Guo Liu 1 Institute for Physical Science andtechnology and Department of Mathe
Convergence of A Galerkin Method for 2-D Discontinuous Euler Flows Jian-Guo Liu 1 Institute for Physical Science andtechnology and Department of Mathematics University of Maryland College Park, MD 2742
More informationarxiv: v1 [math.na] 27 Jan 2016
Virtual Element Method for fourth order problems: L 2 estimates Claudia Chinosi a, L. Donatella Marini b arxiv:1601.07484v1 [math.na] 27 Jan 2016 a Dipartimento di Scienze e Innovazione Tecnologica, Università
More informationQUASI-UNIFORMLY POSITIVE OPERATORS IN KREIN SPACE. Denitizable operators in Krein spaces have spectral properties similar to those
QUASI-UNIFORMLY POSITIVE OPERATORS IN KREIN SPACE BRANKO CURGUS and BRANKO NAJMAN Denitizable operators in Krein spaces have spectral properties similar to those of selfadjoint operators in Hilbert spaces.
More informationA gradient recovery method based on an oblique projection and boundary modification
ANZIAM J. 58 (CTAC2016) pp.c34 C45, 2017 C34 A gradient recovery method based on an oblique projection and boundary modification M. Ilyas 1 B. P. Lamichhane 2 M. H. Meylan 3 (Received 24 January 2017;
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Nonconformity and the Consistency Error First Strang Lemma Abstract Error Estimate
More informationLifting to non-integral idempotents
Journal of Pure and Applied Algebra 162 (2001) 359 366 www.elsevier.com/locate/jpaa Lifting to non-integral idempotents Georey R. Robinson School of Mathematics and Statistics, University of Birmingham,
More informationINTRODUCTION TO FINITE ELEMENT METHODS
INTRODUCTION TO FINITE ELEMENT METHODS LONG CHEN Finite element methods are based on the variational formulation of partial differential equations which only need to compute the gradient of a function.
More informationLinear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) 1.1 The Formal Denition of a Vector Space
Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) Contents 1 Vector Spaces 1 1.1 The Formal Denition of a Vector Space.................................. 1 1.2 Subspaces...................................................
More informationON THE CONSTRUCTION OF DUALLY FLAT FINSLER METRICS
Huang, L., Liu, H. and Mo, X. Osaka J. Math. 52 (2015), 377 391 ON THE CONSTRUCTION OF DUALLY FLAT FINSLER METRICS LIBING HUANG, HUAIFU LIU and XIAOHUAN MO (Received April 15, 2013, revised November 14,
More informationb i (x) u + c(x)u = f in Ω,
SIAM J. NUMER. ANAL. Vol. 39, No. 6, pp. 1938 1953 c 2002 Society for Industrial and Applied Mathematics SUBOPTIMAL AND OPTIMAL CONVERGENCE IN MIXED FINITE ELEMENT METHODS ALAN DEMLOW Abstract. An elliptic
More information1. Introduction. The Stokes problem seeks unknown functions u and p satisfying
A DISCRETE DIVERGENCE FREE WEAK GALERKIN FINITE ELEMENT METHOD FOR THE STOKES EQUATIONS LIN MU, JUNPING WANG, AND XIU YE Abstract. A discrete divergence free weak Galerkin finite element method is developed
More informationC 0 IPG Method for Biharmonic Eigenvalue Problems
C IPG Method for Biharmonic Eigenvalue Problems Susanne C. Brenner, Peter Monk 2, and Jiguang Sun 3 Department of Mathematics and Center for Computation & Technology, Louisiana State University, Baton
More informationProjected Surface Finite Elements for Elliptic Equations
Available at http://pvamu.edu/aam Appl. Appl. Math. IN: 1932-9466 Vol. 8, Issue 1 (June 2013), pp. 16 33 Applications and Applied Mathematics: An International Journal (AAM) Projected urface Finite Elements
More informationLEBESGUE INTEGRATION. Introduction
LEBESGUE INTEGATION EYE SJAMAA Supplementary notes Math 414, Spring 25 Introduction The following heuristic argument is at the basis of the denition of the Lebesgue integral. This argument will be imprecise,
More informationSupraconvergence of a Non-Uniform Discretisation for an Elliptic Third-Kind Boundary-Value Problem with Mixed Derivatives
Supraconvergence of a Non-Uniform Discretisation for an Elliptic Third-Kind Boundary-Value Problem with Mixed Derivatives Etienne Emmrich Technische Universität Berlin, Institut für Mathematik, Straße
More informationLECTURE 1: SOURCES OF ERRORS MATHEMATICAL TOOLS A PRIORI ERROR ESTIMATES. Sergey Korotov,
LECTURE 1: SOURCES OF ERRORS MATHEMATICAL TOOLS A PRIORI ERROR ESTIMATES Sergey Korotov, Institute of Mathematics Helsinki University of Technology, Finland Academy of Finland 1 Main Problem in Mathematical
More informationContents Introduction Energy nalysis of Convection Equation 4 3 Semi-discrete nalysis 6 4 Fully Discrete nalysis 7 4. Two-stage time discretization...
Energy Stability nalysis of Multi-Step Methods on Unstructured Meshes by Michael Giles CFDL-TR-87- March 987 This research was supported by ir Force Oce of Scientic Research contract F4960-78-C-0084, supervised
More informationNumerische Mathematik
Numer. Math. (997) 76: 479 488 Numerische Mathematik c Springer-Verlag 997 Electronic Edition Exponential decay of C cubic splines vanishing at two symmetric points in each knot interval Sang Dong Kim,,
More informationComputers and Mathematics with Applications. Convergence analysis of the preconditioned Gauss Seidel method for H-matrices
Computers Mathematics with Applications 56 (2008) 2048 2053 Contents lists available at ScienceDirect Computers Mathematics with Applications journal homepage: wwwelseviercom/locate/camwa Convergence analysis
More informationLOW ORDER CROUZEIX-RAVIART TYPE NONCONFORMING FINITE ELEMENT METHODS FOR APPROXIMATING MAXWELL S EQUATIONS
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 5, Number 3, Pages 373 385 c 28 Institute for Scientific Computing and Information LOW ORDER CROUZEIX-RAVIART TYPE NONCONFORMING FINITE ELEMENT
More informationRealization of set functions as cut functions of graphs and hypergraphs
Discrete Mathematics 226 (2001) 199 210 www.elsevier.com/locate/disc Realization of set functions as cut functions of graphs and hypergraphs Satoru Fujishige a;, Sachin B. Patkar b a Division of Systems
More informationNP-hardness of the stable matrix in unit interval family problem in discrete time
Systems & Control Letters 42 21 261 265 www.elsevier.com/locate/sysconle NP-hardness of the stable matrix in unit interval family problem in discrete time Alejandra Mercado, K.J. Ray Liu Electrical and
More informationOverview. A Posteriori Error Estimates for the Biharmonic Equation. Variational Formulation and Discretization. The Biharmonic Equation
Overview A Posteriori rror stimates for the Biharmonic quation R Verfürth Fakultät für Mathematik Ruhr-Universität Bochum wwwruhr-uni-bochumde/num1 Milan / February 11th, 013 The Biharmonic quation Summary
More informationMODIFIED STREAMLINE DIFFUSION SCHEMES FOR CONVECTION-DIFFUSION PROBLEMS. YIN-TZER SHIH AND HOWARD C. ELMAN y
MODIFIED STREAMLINE DIFFUSION SCHEMES FOR CONVECTION-DIFFUSION PROBLEMS YIN-TZER SHIH AND HOWARD C. ELMAN y Abstract. We consider the design of robust and accurate nite element approximation methods for
More informationA mixed finite element approximation of the Stokes equations with the boundary condition of type (D+N)
wwwijmercom Vol2, Issue1, Jan-Feb 2012 pp-464-472 ISSN: 2249-6645 A mixed finite element approximation of the Stokes equations with the boundary condition of type (D+N) Jaouad El-Mekkaoui 1, Abdeslam Elakkad
More informationSpurious Chaotic Solutions of Dierential. Equations. Sigitas Keras. September Department of Applied Mathematics and Theoretical Physics
UNIVERSITY OF CAMBRIDGE Numerical Analysis Reports Spurious Chaotic Solutions of Dierential Equations Sigitas Keras DAMTP 994/NA6 September 994 Department of Applied Mathematics and Theoretical Physics
More informationA fourth-order nite-dierence method based on non-uniform mesh for a class of singular two-point boundary value problems
Journal of Computational and Applied Mathematics 136 (2001) 337 342 www.elsevier.com/locate/cam A fourth-order nite-dierence method based on non-uniform mesh for a class of singular two-point boundary
More informationINTERGRID OPERATORS FOR THE CELL CENTERED FINITE DIFFERENCE MULTIGRID ALGORITHM ON RECTANGULAR GRIDS. 1. Introduction
Trends in Mathematics Information Center for Mathematical Sciences Volume 9 Number 2 December 2006 Pages 0 INTERGRID OPERATORS FOR THE CELL CENTERED FINITE DIFFERENCE MULTIGRID ALGORITHM ON RECTANGULAR
More informationarxiv: v1 [math.na] 27 Jan 2016
Virtual Element Method for fourth order problems: L 2 estimates Claudia Chinosi a, L. Donatella Marini b arxiv:1601.07484v1 [math.na] 27 Jan 2016 a Dipartimento di Scienze e Innovazione Tecnologica, Università
More informationMonte Carlo Methods for Statistical Inference: Variance Reduction Techniques
Monte Carlo Methods for Statistical Inference: Variance Reduction Techniques Hung Chen hchen@math.ntu.edu.tw Department of Mathematics National Taiwan University 3rd March 2004 Meet at NS 104 On Wednesday
More informationAn additive average Schwarz method for the plate bending problem
J. Numer. Math., Vol. 10, No. 2, pp. 109 125 (2002) c VSP 2002 Prepared using jnm.sty [Version: 02.02.2002 v1.2] An additive average Schwarz method for the plate bending problem X. Feng and T. Rahman Abstract
More informationJournal of Computational and Applied Mathematics. Multigrid method for solving convection-diffusion problems with dominant convection
Journal of Computational and Applied Mathematics 226 (2009) 77 83 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
More informationOn a kind of restricted edge connectivity of graphs
Discrete Applied Mathematics 117 (2002) 183 193 On a kind of restricted edge connectivity of graphs Jixiang Meng a; ;1, Youhu Ji b a Department of Mathematics, Xinjiang University, Urumqi 830046, Xinjiang,
More informationElement diameter free stability parameters. for stabilized methods applied to uids
Element diameter free stability parameters for stabilized methods applied to uids by Leopoldo P. Franca Laboratorio Nacional de Computac~ao Cientica (LNCC/CNPq) Rua Lauro Muller 455 22290 Rio de Janeiro,
More informationResearch Article A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations
Abstract and Applied Analysis Volume 212, Article ID 391918, 11 pages doi:1.1155/212/391918 Research Article A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations Chuanjun Chen
More informationPriority Program 1253
Deutsche Forschungsgemeinschaft Priority Program 1253 Optimization with Partial Differential Equations Klaus Deckelnick and Michael Hinze A note on the approximation of elliptic control problems with bang-bang
More informationDisplacement rank of the Drazin inverse
Available online at www.sciencedirect.com Journal of Computational and Applied Mathematics 167 (2004) 147 161 www.elsevier.com/locate/cam Displacement rank of the Drazin inverse Huaian Diao a, Yimin Wei
More informationConvergence of The Multigrid Method With A Wavelet. Abstract. This new coarse grid operator is constructed using the wavelet
Convergence of The Multigrid Method With A Wavelet Coarse Grid Operator Bjorn Engquist Erding Luo y Abstract The convergence of the two-level multigrid method with a new coarse grid operator is studied.
More informationA general theory of discrete ltering. for LES in complex geometry. By Oleg V. Vasilyev AND Thomas S. Lund
Center for Turbulence Research Annual Research Briefs 997 67 A general theory of discrete ltering for ES in complex geometry By Oleg V. Vasilyev AND Thomas S. und. Motivation and objectives In large eddy
More informationOn the equivalence of regularity criteria for triangular and tetrahedral finite element partitions
Computers and Mathematics with Applications 55 (2008) 2227 2233 www.elsevier.com/locate/camwa On the equivalence of regularity criteria for triangular and tetrahedral finite element partitions Jan Brandts
More informationRening the submesh strategy in the two-level nite element method: application to the advection diusion equation
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2002; 39:161 187 (DOI: 10.1002/d.219) Rening the submesh strategy in the two-level nite element method: application to
More informationt x 0.25
Journal of ELECTRICAL ENGINEERING, VOL. 52, NO. /s, 2, 48{52 COMPARISON OF BROYDEN AND NEWTON METHODS FOR SOLVING NONLINEAR PARABOLIC EQUATIONS Ivan Cimrak If the time discretization of a nonlinear parabolic
More informationASYMPTOTICALLY EXACT A POSTERIORI ESTIMATORS FOR THE POINTWISE GRADIENT ERROR ON EACH ELEMENT IN IRREGULAR MESHES. PART II: THE PIECEWISE LINEAR CASE
MATEMATICS OF COMPUTATION Volume 73, Number 246, Pages 517 523 S 0025-5718(0301570-9 Article electronically published on June 17, 2003 ASYMPTOTICALLY EXACT A POSTERIORI ESTIMATORS FOR TE POINTWISE GRADIENT
More informationLocal flux mimetic finite difference methods
Local flux mimetic finite difference methods Konstantin Lipnikov Mikhail Shashkov Ivan Yotov November 4, 2005 Abstract We develop a local flux mimetic finite difference method for second order elliptic
More informationOptimal blocking of two-level fractional factorial designs
Journal of Statistical Planning and Inference 91 (2000) 107 121 www.elsevier.com/locate/jspi Optimal blocking of two-level fractional factorial designs Runchu Zhang a;, DongKwon Park b a Department of
More information