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Superconvergence and Asymptotic Expansions for Bilinear Finite Volume Element Approximations

Published online by Cambridge University Press:  28 May 2015

Cunyun Nie ,
Shi Shu ,
Haiyuan Yu  and
Juan Wu
Cunyun Nie*
Affiliation:
Department of Mathematics and Physics, Hunan Institute of Engineering, Hunan 411104, China
Shi Shu*
Affiliation:
Hunan Key Laboratory for Computation & Simulation in Science and Engineering and Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education, Xiangtan University, Hunan 411105, China
Haiyuan Yu*
Affiliation:
Hunan Key Laboratory for Computation & Simulation in Science and Engineering and Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education, Xiangtan University, Hunan 411105, China
Juan Wu*
Affiliation:
Hunan Key Laboratory for Computation & Simulation in Science and Engineering and Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education, Xiangtan University, Hunan 411105, China
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
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Abstract

Aiming at the isoparametric bilinear finite volume element scheme, we initially derive an asymptotic expansion and a high accuracy combination formula of the derivatives in the sense of pointwise by employing the energy-embedded method on uniform grids. Furthermore, we prove that the approximate derivatives are convergent of order two. Finally, numerical examples verify the theoretical results.

Keywords

65M1065M0841A60Isoparametric bilinear finite volume element schemeasymptotic expansionhigh accuracy combination formulasuperconvergence
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 6 , Issue 2 , May 2013 , pp. 408 - 423
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Li, R. H. and Zhu, P. Q., Generalized difference methods for second order elliptic partial differential equations quadrilateral grids II, Numer. Math. J. Chinese Univ., 4 (1982), pp. 360–375.Google Scholar
[2]Schmidt, T., Box schemes on quadrilateral meshes, Computing, 51 (1993), pp. 271–292.CrossRefGoogle Scholar
[3]Porsching, T. A., Error estimates for MAC-like approximations to the linear Navier-Stokes equations, Numer. Math., 29 (1987), pp. 291–364.CrossRefGoogle Scholar
[4]Chou, S. H., Analysis and convergence of a covolume element method for the generalized Stokes problem, Math. Comput., 66 (1997), pp. 85–104.CrossRefGoogle Scholar
[5]Rui, H. X., Symmetric modified finite volume element methods for self-adjoint elliptic and parabolic problems, J. Comput. Appl. Math., 146 (2002), pp. 373–386.CrossRefGoogle Scholar
[6]Ma, X., Shu, S. and Zhou, A., Symmetric finite volume discretization for parabolic problems, Comput. Meth. Appl. Mech. Eng., 192 (2003), pp. 4467–4485.CrossRefGoogle Scholar
[7]Cai, Z. Q., Douglas, J. and Park, M., Development and analysis of higher order finite volume methods over rectangles for elliptic equations, Adv. Comput. Math., 19 (2003), pp. 3–33.CrossRefGoogle Scholar
[8]Yang, M., Cubic finite volume methods for second order elliptic equations with variable coefi-cients, Northeastern Mathematical Journality, 21 (2005), pp. 146–152.Google Scholar
[9]Chen, L., A new class of high order finite volume methods for second order elliptic equations, SIAM. J. Numer. Anal., 47 (2010), pp. 4021–4043.CrossRefGoogle Scholar
[10]Wang, T. K., Alternating direction finite volume element methods for three-dimensional parabolic equations, Numer. Math. Theor. Meth. Appl., 3 (2010), pp. 499–522.CrossRefGoogle Scholar
[11]Abedini, A. A. and Ghiassi, R. A., Three-dimensional finite volume model for shallow water flow simulation, Ausrtalian J. Basic AN., 4 (2010), pp. 3208–3215.Google Scholar
[12]Chen, Z. Y., Li, R. H. and Zhou, A. H., A note on the optimal L2-estimate of the finite volume element method, Adv. Comput. Math., 16 (2002), pp. 291–303.CrossRefGoogle Scholar
[13]Lv, J. L. and Y Li, H., em L 2 error estimate of the finite volume element methods on quadrilateral meshes, Adv. Comput. Math., 33 (2010), pp. 129–148.CrossRefGoogle Scholar
[14]Li, Y. H. and Li, R. H., Generalized difference methods on arbitrary quadrilateral networks, J. Comput. Math., 6 (1999), pp. 653–672.Google Scholar
[15]Lv, J.L., L2 Error Estimates and Superconvergence of the Finite Volume Element Methods on Quadrilaterial Meshes (in Chinese), JiLin University Doctor thesis, 2009.Google Scholar
[16]Chen, C. M. and Huang, Y. Q., High Accuracy of Finite Element Method (in chinese), Hunan Science Press, China, 1995.Google Scholar
[17]Chen, C. M., Structure Theory of Superconvergence of Finite Elements (in chinese), Hunan Science Press, China, 2001.Google Scholar
[18]Huang, Y. Q., Qin, H. F. and Wang, D. S., Centroidal Voronoi Tessellation-based finite element superconvergence, Int. J. Numer. Methods Eng., 76 (2008), pp. 1819–1839.CrossRefGoogle Scholar
[19]Süli, E., Convergence of finite volume schemes for Poisson’s equation on nonuniform meshes, Int. J. Numer. Anal. Mod., 3 (2006), pp. 348–360.Google Scholar
[20]Shu, S., Yu, H. Y., Huang, Y. Q. and Nie, C. Y., A preserving-symmetry finite volume scheme and superconvergence on quadrangle grids, Int. J. Numer. Anal. Mod., 3 (2006), pp. 348–360.Google Scholar
[21]Nie, C. Y., Several Finite Volume Element Schemes and Some Applications in Radiation Heat Conduction Problems (in Chinese), Xiangtan University Doctor thesis, 2010.Google Scholar
[22]Zhang, L., On convergence of isoparametric bilinear finite elements, Commun. Numer. Meth. Eng., 12 (1996), pp. 849–862.3.0.CO;2-N>CrossRefGoogle Scholar