Postprocessing of a finite volume element method for semilinear parabolic problems

Yang, Min; Bi, Chunjia; Liu, Jiangguo

doi:10.1051/m2an/2009017

Yang, Min ; Bi, Chunjia ; Liu, Jiangguo ¹

¹ Department of Mathematics, Colorado State University, Fort Collins, CO 80523-1874, USA.

ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 5, pp. 957-971.

Résumé

In this paper, we study a postprocessing procedure for improving accuracy of the finite volume element approximations of semilinear parabolic problems. The procedure amounts to solve a source problem on a coarser grid and then solve a linear elliptic problem on a finer grid after the time evolution is finished. We derive error estimates in the $L^{2}$ and $H^{1}$ norms for the standard finite volume element scheme and an improved error estimate in the $H^{1}$ norm. Numerical results demonstrate the accuracy and efficiency of the procedure.

MR Zbl

DOI : 10.1051/m2an/2009017

Classification : 65N30, 65N15
Mots clés : error estimates, finite volume elements, postprocessing, semilinear parabolic problems

Affiliations des auteurs :

Yang, Min ; Bi, Chunjia ; Liu, Jiangguo ¹

¹ Department of Mathematics, Colorado State University, Fort Collins, CO 80523-1874, USA.

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     author = {Yang, Min and Bi, Chunjia and Liu, Jiangguo},
     title = {Postprocessing of a finite volume element method for semilinear parabolic problems},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {957--971},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {5},
     year = {2009},
     doi = {10.1051/m2an/2009017},
     mrnumber = {2559740},
     zbl = {1176.65102},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2009017/}
}

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Yang, Min; Bi, Chunjia; Liu, Jiangguo. Postprocessing of a finite volume element method for semilinear parabolic problems. ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 5, pp. 957-971. doi : 10.1051/m2an/2009017. http://archive.numdam.org/articles/10.1051/m2an/2009017/

Bibliographie
Cité par

[1] R. Adams and J.J.F. Fournier, Sobolev Spaces. Academic Press, New York (2003). | MR | Zbl

[2] C. Bi and V. Ginting, Two-grid finite volume element method for linear and nonlinear elliptic problems. Numer. Math. 107 (2007) 177-198. | MR | Zbl

[3] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York, 2nd edn., (2002). | MR | Zbl

[4] Z. Cai, On the finite volume element method. Numer. Math. 58 (1991) 713-735. | MR | Zbl

[5] Z. Cai, J. Mandel and S. Mccormick, The finite volume element method for diffusion equations on general triangulations. SIAM J. Numer. Anal. 28 (1991) 392-402. | MR | Zbl

[6] C. Carstensen, R. Lazarov and S. Tomov, Explicit and averaging a posteriori error estimates for adaptive finite volume methods. SIAM J. Numer. Anal. 42 (2005) 2496-2521. | MR | Zbl

[7] P. Chatzipantelidis and R.D. Lazarov, Error estimates for a finite volume element method for elliptic PDEs in nonconvex polygonal domains. SIAM J. Numer. Anal. 42 (2004) 1932-1958. | MR | Zbl

[8] P. Chatzipantelidis, R.D. Lazarov and V. Thomée, Error estimates for a finite volume element method for parabolic equations in convex polygonal domains. Numer. Meth. PDEs 20 (2004) 650-674. | MR | Zbl

[9] S.H. Chou and D.Y. Kwak, Multigrid algorithms for a vertex-centered covolume method for elliptic problems. Numer. Math. 90 (2002) 459-486. | MR | Zbl

[10] S.H. Chou and Q. Li, Error estimates in $L^{2}$ , $H^{1}$ and $L^{\infty}$ in covolume methods for elliptic and parabolic problems: a unified approach. Math. Comp. 69 (2000) 103-120. | MR | Zbl

[11] S.H. Chou, D.Y. Kwak and Q. Li, $L^{p}$ error estimates and superconvergence for covolume or finite volume element methods. Numer. Meth. PDEs 19 (2003) 463-486. | MR | Zbl

[12] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | MR | Zbl

[13] C.N. Dawson, M.F. Wheeler and C.S. Woodward, A two-grid finite difference scheme for nonlinear parabolic equations. SIAM J. Numer. Anal. 35 (1998) 435-452. | MR | Zbl

[14] J. De Frutos and J. Novo, Postprocessing the linear finite element method. SIAM J. Numer. Anal. 40 (2002) 805-819. | MR | Zbl

[15] R.E. Ewing, T. Lin and Y. Lin, On the accuracy of the finite volume element method based on piecewise linear polynomials. SIAM J. Numer. Anal. 39 (2002) 1865-1888. | MR | Zbl

[16] R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods: Handbook of Numerical Analysis. North-Holland, Amsterdam (2000). | MR | Zbl

[17] M. Feistauer, J. Felcman, M. Lukáčová-Medvidová and G. Warnecke, Error estimates of a combined finite volume-finite element method for nonlinear convection-diffusion problems. SIAM J. Numer. Anal. 36 (1999) 1528-1548. | MR | Zbl

[18] B. García-Archilla, J. Novo and E.S. Titi, Postprocessing the Galerkin method: a novel approach to approximate inertial manifolds. SIAM J. Numer. Anal. 35 (1998) 941-972. | MR | Zbl

[19] B. García-Archilla and E.S. Titi, Postprocessing the Galerkin method: the finite element case. SIAM J. Numer. Anal. 37 (2000) 470-499. | MR | Zbl

[20] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics 840. Springer-Verlag, New York (1989). | MR | Zbl

[21] A. Lasis and E. Süli, $h p$ -version discontinuous Galerkin finite element method for semilinear parabolic problems. SIAM J. Numer. Anal. 45 (2007) 1544-1569. | MR | Zbl

[22] R. Li, Z. Chen and W. Wu, Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods. Marcel Dekker, New York (2000). | MR | Zbl

[23] X. Ma, S. Shu and A. Zhou, Symmetric finite volume discretizations for parabolic problems. Comput. Methods Appl. Mech. Engrg. 192 (2003) 4467-4485. | MR | Zbl

[24] M. Marion and J.C. Xu, Error estimates on a new nonlinear Galerkin method based on two-grid finite elements. SIAM J. Numer. Anal. 32 (1995) 1170-1184. | MR | Zbl

[25] H. Rui, Symmetric modified finite volume element methods for self-adjoint elliptic and parabolic problems. J. Comput. Appl. Math. 146 (2002) 373-386. | MR | Zbl

[26] A.H. Schatz, V. Thomée and L. Wahlbin, Maximum norm stability and error estimates in parabolic finite element equations. Comm. Pure Appl. Math. 33 (1980) 265-304. | MR | Zbl

[27] R.K. Sinha and J. Geiser, Error estimates for finite volume element methods for convection-diffusion-reaction equations. Appl. Numer. Math. 57 (2007) 59-72. | MR | Zbl

[28] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences 68. Springer-Verlag, Berlin (1988). | MR | Zbl

[29] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, Berlin (1997). | MR | Zbl

[30] V. Thomée and L. Wahlbin, On Galerkin methods in semilinear parabolic problems. SIAM J. Numer. Anal. 12 (1975) 378-389. | MR | Zbl

[31] Y. Yan, Postprocessing the finite element method for semilinear parabolic problems. SIAM J. Numer. Anal. 44 (2006) 1681-1702. | MR | Zbl

[32] M. Yang, A second-order finite volume element method on quadrilateral meshes for elliptic equations. ESAIM: M2AN 40 (2006) 1053-1067. | EuDML | Numdam | MR | Zbl

[33] X. Ye, A discontinuous finite volume method for the Stokes problems. SIAM J. Numer. Anal. 44 (2006) 183-198. | MR | Zbl

[34] S. Zhang, On domain decomposition algorithms for covolume methods for elliptic problems. Comput. Methods Appl. Mech. Engrg. 196 (2006) 24-32. | MR | Zbl

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