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A Quadratic Triangular Finite Volume Element Method for a Semilinear Elliptic Equation

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  11 October 2016

Zhiguang Xiong  and
Kang Deng
Zhiguang Xiong*
Affiliation:
School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China
Kang Deng*
Affiliation:
School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China
*
*Corresponding author. Email:zgxiong@hnust.edu.cn (Z. G. Xiong), kdeng@hnust.edu.cn (K. Deng)
*Corresponding author. Email:zgxiong@hnust.edu.cn (Z. G. Xiong), kdeng@hnust.edu.cn (K. Deng)
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Abstract

In this paper we extend the idea of interpolated coefficients for a semilinear problem to the quadratic triangular finite volume element method. At first we introduce quadratic triangular finite volume element method with interpolated coefficients for a boundary value problem of semilinear elliptic equation. Next we derive convergence estimate in H1-norm, L2-norm and L∞-norm, respectively. Finally an example is given to illustrate the effectiveness of the proposed method.

Keywords

Semilinear elliptic equationtriangulationfinite volume element with interpolated coefficients

MSC classification

Secondary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 1 , February 2017 , pp. 186 - 204
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Adams, R. A., Sobolev Spaces, Pure and Applied Mathematics, Vol. 65, Academic Press (A subsidiary of Harcourt Brace Jovanovich, Publishers, New York, 1975.Google Scholar
[2] Bank, R. E. and Rose, D. J., Some error estimates for the box method, SIAM J. Numer. Anal., 24 (1987), pp. 777–787.Google Scholar
[3] Bi, C. J., Superconvergence of finite volume element method for a nonlinear elliptic problem, Numer. Methods PDEs, 23 (2007), pp. 220–233.CrossRefGoogle Scholar
[4] Bramble, J. H., A second-order finite difference analog of the first biharmonia boundary value problem, Numer. Math., 9(4) (1966), pp. 236–249.CrossRefGoogle Scholar
[5] Cai, Z. Q., On the finite volume element method, Numer. Math., 58 (1991), pp. 713–735.Google Scholar
[6] Cai, Z. Q. and McCormick, Steve, On the accuracy of the finite volume element method for diffusion equations on composite grids, SIAM J. Numer. Anal., 27 (1990), pp. 636–655.Google Scholar
[7] Chatzipantelidis, P., Finite volume methods for elliptic PDE’s: a new approach, M2AN Math. Model. Numer. Anal., 3 (2002), pp. 307–324.Google Scholar
[8] Chatzipantelidis, P., Ginting, V. and Lazarov, R. D., A finite volume element for a non-linear elliptic problem, Numer. Linear Algebra Appl., 12 (2005), pp. 515–546.Google Scholar
[9] Chen, C. M. and Huang, Y. Q., High Accuracy Theory of Finite Element Methods, Hunan Science and Technology Publishers, 1995, in Chinese.Google Scholar
[10] Chen, C. M., Larson, S. and Zhang, N. Y., Error estimates of optimal order for finite element methods interpolated coefficients for the nonlinear heat equation, IMA J. Numer. Anal., 9 (1989), pp. 507–524.CrossRefGoogle Scholar
[11] Chou, S. H., Kwak, D. Y. and Li, Q., Lp error estimates and superconvergence for covolume or finite volume element methods, Numer. Methods PDEs, 19 (2003), pp. 463–486.Google Scholar
[12] Chou, S. H. and Li, Q., Error estimates in L2, H1 and L∞ in covolume methods for elliptic and parabolic problems: a unified approach, Math. Comput., 69 (2000), pp. 103–120.Google Scholar
[13] Douglas, J. Jr and Dupont, T., A Galerkin method for a nonlinear Dirichlet problem, Math. Comput., 29 (1975), pp. 689–696.Google Scholar
[14] Ewing, R. E, Lazarov, R. and Lin, Y., Finite volume element approximations of nonlocal reactive flows in porous media, Numer. Methods PDEs, 16 (2000), pp. 285–311.Google Scholar
[15] Ewing, R. E., Lin, T. and Lin, Y. P., On the accuracy of the finite volume element method based on piecewise linear polynomials, SIAM J. Numer. Anal., 39 (2002), pp. 1865–1888.Google Scholar
[16] Frehse, J. and Rannacher, R., Asymptotic L∞-error estimate for linear finite element appproximations of quasilinear boundary value problems, SIAM J. Mumer. Anal., 15 (1978), pp. 418–431.CrossRefGoogle Scholar
[17] Huang, J. G. and Li, L. K., Some superconvergence results for the covolume method for elliptic problems, Commun. Numer. Meth. Eng., 17 (2001), pp. 291–302.Google Scholar
[18] Huang, J. G. and Xi, S. T., On the finite volume element method for general self-adjoint elliptic problems, SIAM J. Numer. Anal., 35 (1998), pp. 1762–1774.Google Scholar
[19] Larson, S., Thomee, V. and Zhang, N. Y., Interpolation of coefficients and transformation of dependent variable in element methods for the nonlinear heat equation, Math. Methods Appl. Sci., 11 (1989), pp. 105–124.CrossRefGoogle Scholar
[20] Li, R. H., Generalized differential methods for a nonlinear Dirichlet problem, SIAM J. Numer. Anal., 24 (1987), pp. 77–88.Google Scholar
[21] Li, R. H., Chen, Z. Y. and Wu, W., Generalized Difference Methods for Differential Equations, volume 226 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker Inc, New York, 2000, Numerical Analysis of Finite Volume Methods.Google Scholar
[22] Li, Y. H. and Li, R. H., Generalized diffierence methods on arbitrary quadrilateral networks, J. Comput. Math., 17(6) (1999).Google Scholar
[23] Matsuzawa, Y., Finite element approximation for some quasilinear elliptic problems, J. Comput. Appl. Math., 96 (1998), pp. 13–25.Google Scholar
[24] Nochetto, R. H., Pointwise a posteriori error estimates for elliptic problems on highly graded meshes, Math. Comput., 64 (1995), pp. 1–22.CrossRefGoogle Scholar
[25] Selmin, V., The node-centred finite volume approach: Bridge between finite differences and finite elements, Comput. Methods Appl. Mech. Eng., 102 (1993), pp. 107–138.Google Scholar
[26] Suli, E., Convergence of finite volume schemes for Poissons equation on nonuniform meshes, SIAM J. Numer. Anal., 28 (1991), pp. 1419–1430.CrossRefGoogle Scholar
[27] Suli, E., The accuracy of cell vertex finite volume methods on quadrilateral meshes, Math. Comput., 59 (1992), pp. 359–382.CrossRefGoogle Scholar
[28] Wu, H. J. and Li, R. H., Error estimates for finite volume element methods for general second-order elliptic problem, Numer. Methods PDEs, 19 (2003), pp. 693–708.Google Scholar
[29] Xiong, Z. G. and Chen, C. M., Superconvergence of triangular quadratic finite element method with interpolated coefficients for nonlinear elliptic problem, Acta Mathematics Scientia, Ser. A, 26(2) (2006), pp. 174–182.Google Scholar
[30] Xiong, Z. G. and Chen, C. M., Superconvergence of rectangular finite element with interpolated coefficients for semilinear elliptic problem, Appl. Math. Comput., 181 (2006), pp. 1577–1584.Google Scholar
[31] Xiong, Z. G. and Chen, C. M., Superconvergence of continuous finite elements with interpolated coefficients for initial value problems of nonlinear ordinary differential equation, Numerical Mathematics: A Journal of Chinese Universities, (English Ser.), 16(1) (2007), pp. 37–44.Google Scholar
[32] Xiong, Z. G. and Chen, Y. P., Finite volume element method with interpolated coefficients for two-point boundary value problem of semilinear differential equations, Comput. Methods Appl. Mech. Eng., 196 (2007), pp. 3798–3804.Google Scholar
[33] Xiong, Z. G. and Chen, Y. P., Triangular finite volume element method with interpolated coefficients for a semilinear elliptic equation, J. Comput. Math., 32(2) (2014), pp. 152–168.Google Scholar
[34] Zhu, P. and Li, R., Generalized difference methods for second order elliptic partial differential equations (II)-quadrilateral grids, Numerical Mathematics: A Journal of Chinese Universities, 4 (1982), pp. 360–375.Google Scholar
[35] Zlamal, M., A finite element solution of the nonlinear heat equation, RAIRO Model. Anal. Numer., 14 (1980), pp. 203–216.CrossRefGoogle Scholar