×

zbMATH — the first resource for mathematics

Superconvergence of finite volume methods for the Stokes equations. (English) Zbl 1170.76037
Summary: A general superconvergence result for finite volume method for Stokes equations is obtained by using a \(L^2\) projection post-processing technique. This superconvergence result can be applied to different finite volume methods and to general quasi-uniform meshes.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
PDF BibTeX Cite
Full Text: DOI
References:
[1] Douglas, A superconvergence for mixed finite element methods on rectangular domains, Calcolo 26 pp 121– (1989) · Zbl 0714.65084
[2] Ewing, Superconvergence of mixed finite element approximations over quadrilaterals, SIAM J Numer Anal 36 pp 772– (1998) · Zbl 0926.65107
[3] Wahlbin 1605 (1995)
[4] Schatz, Superconvergence in finite element methods and meshes that are symmetric with respect to a point, SIAM J Numer Anal 33 pp 505– (1996) · Zbl 0855.65115
[5] Lin, Analysis and construction of finite element methods with high efficiency (1996)
[6] Li, Analysis of a class of superconvergence patch recovery techniques for linear and bilinear finite elements, Numer Methods Partial Differential Eq 15 pp 151– (1999)
[7] Wang, Superconvergence of finite element approximations for the Stokes problem by least squares surface fitting, SIAM J Numer Anal 39 pp 1001– (2001)
[8] Zienkiewicz, The superconvergent patch recovery and a posteriori error estimates. I. The recovery technique, Int J Numer Methods Eng 33 pp 1331– (1992) · Zbl 0769.73084
[9] Zienkiewicz, The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity, Int. J Numer Methods Eng 33 pp 1365– (1992) · Zbl 0769.73085
[10] Zhang, Analysis of the superconvergent patch recovery technique and a posteriori error estimator in the finite element method (II), Comput Methods Appl Mech Eng 163 pp 159– (1998) · Zbl 0941.65116
[11] Zlamal, Superconvergence and reduced integration in the finite element method, Math Comp 32 pp 663– (1977) · Zbl 0424.65059
[12] Cai, The finite volume element method for diffusion equations on general triangulations, SIAM J Numer Anal 28 pp 392– (1991) · Zbl 0729.65086
[13] Cai, On the accuracy of the finite volume element method for diffusion equations on composite grids, SIAM J Numer Anal 27 pp 636– (1990) · Zbl 0707.65073
[14] Chou, Analysis and convergence of a covolume method for the generalized Stokes problem, Math Comp 217 pp 85– (1997) · Zbl 0854.65091
[15] Chou, A covolume method based on rotated bilinears for the generalized Stokes problem, SIAM J Numer Anal 2 pp 494– (1998) · Zbl 0957.76027
[16] Chou, Error estimates in L2, H1 and L in covolume methods for elliptic and parabolic problems, a unified approach, Math Comp 69 pp 103– (2000)
[17] Chou, A general mixed co-volume framework for constructing conservative schemes for elliptic problems, Math Comp 68 pp 991– (1999)
[18] Chou, Unified analysis of finite volume methods for second order elliptic problems, SIAM Numer Anal 45 pp 1639– (2007) · Zbl 1155.65099
[19] Ewing, On the accuracy of the finite volume element method based on piecewise linear polynomials, SIAM J Numer Anal 39 pp 1865– (2002) · Zbl 1036.65084
[20] Lazarov, Finite volume methods for convection-diffusion problems, SIAM J Numer Anal 33 pp 31– (1996) · Zbl 0847.65075
[21] Li, A new stabilized finite volume method for the stationary Stokes equations, Adv Comput Mathe
[22] Li, Generalized difference methods for two point boundary value problems, em Actu Sci Natur Univ Jilin 1 pp 26– (1982)
[23] Li, Generalized difference methods for a nonlinear Dirichlet problem, SIAM J Numer Math 24 pp 77– (1987) · Zbl 0626.65091
[24] Li, Generalized difference methods for second order elliptic partial differential equations (1): triangle grid, Numer Math J Chinese Univ 2 pp 140– (1982)
[25] Li, The generalized difference method for differential equations (1994)
[26] Li, Generalized difference methods for differential equations-Numerical analysis of finite volume methods, Monographs and Textbooks in Pure and Applied Mathematics 226 (2000)
[27] Ye, A new discontinuous finite volume method for elliptic problems, SIAM J Numer Anal 42 pp 1062– (2004) · Zbl 1079.65116
[28] Ye, A discontinuous finite volume method for the Stokes problem, SIAM J Numer Anal 44 pp 183– (2006) · Zbl 1112.65125
[29] Chen, L2 estimates of linear element generalized difference schemes, Actu Sci Natur Univ Sunyaseni 33 pp 24– (1994)
[30] Chen, A note on the optimal L2-estimat of the finite valume element method, Adv Comp Math 16 pp 291– (2002)
[31] Chatzipantelidis, Finite volume methods for elliptic PDE’s: a new approach, Math Model Numer Anal 36 pp 307– (2002) · Zbl 1041.65087
[32] Jianguo, On the finite volume element method for general self-adjoint problems, SIAM J Numer Anal 35 pp 1762– (1998) · Zbl 0913.65097
[33] Wang, A superconvergence analysis for finite element solutions by the least-squares surface fitting on irregular meshes for smooth problems, J Math Study 33 pp 229– (2000) · Zbl 0987.65108
[34] Arnold, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J Numer Anal 39 pp 1749– (2002) · Zbl 1008.65080
[35] Cui, Unified analysis of finite volume methods for the Stokes equations, SIAM Numer Anal
[36] Ye, On the relationship between finite volume and finite element methods applied to the Stokes equations, Numer Methods Partial Differential Eq 17 pp 440– (2001) · Zbl 1017.76057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.