Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-24T13:33:23.295Z Has data issue: false hasContentIssue false
  • English
  • Français

Postprocessing of a finite volume element methodfor semilinear parabolic problems

Published online by Cambridge University Press:  12 June 2009

Min Yang ,
Chunjia Bi  and
Jiangguo Liu
Min Yang
Affiliation:
Department of Mathematics, Yantai University, Yantai, Shandong 264005, P. R. China. [email protected]; [email protected]
Chunjia Bi
Affiliation:
Department of Mathematics, Yantai University, Yantai, Shandong 264005, P. R. China. [email protected]; [email protected]
Jiangguo Liu
Affiliation:
Department of Mathematics, Colorado State University, Fort Collins, CO 80523-1874, USA. [email protected]
Get access

Abstract

In this paper, we study a postprocessing procedure for improvingaccuracy of the finite volume element approximations of semilinearparabolic problems. The procedure amounts to solve a source problemon a coarser grid and then solve a linear elliptic problem on afiner grid after the time evolution is finished. We derive errorestimates in the L 2 and H 1 norms for the standard finitevolume element scheme and an improved error estimate in the H 1 norm. Numerical results demonstrate the accuracy and efficiency ofthe procedure.

Keywords

Error estimatesfinite volume elementspostprocessingsemilinear parabolic problems
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 5 , September 2009 , pp. 957 - 971
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R. Adams and J.J.F. Fournier, Sobolev Spaces. Academic Press, New York (2003).
Bi, C. and Ginting, V., Two-grid finite volume element method for linear and nonlinear elliptic problems. Numer. Math. 107 (2007) 177–198. CrossRef
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York, 2nd edn., (2002).
Cai, Z., On the finite volume element method. Numer. Math. 58 (1991) 713–735. CrossRef
Cai, Z., Mandel, J. and McCormick, S., The finite volume element method for diffusion equations on general triangulations. SIAM J. Numer. Anal. 28 (1991) 392–402. CrossRef
Carstensen, C., Lazarov, R. and Tomov, S., Explicit and averaging a posteriori error estimates for adaptive finite volume methods. SIAM J. Numer. Anal. 42 (2005) 2496–2521. CrossRef
Chatzipantelidis, P. and Lazarov, R.D., Error estimates for a finite volume element method for elliptic PDEs in nonconvex polygonal domains. SIAM J. Numer. Anal. 42 (2004) 1932–1958. CrossRef
Chatzipantelidis, P., Lazarov, R.D. and Thomée, V., Error estimates for a finite volume element method for parabolic equations in convex polygonal domains. Numer. Meth. PDEs 20 (2004) 650–674. CrossRef
Chou, S.H. and Kwak, D.Y., Multigrid algorithms for a vertex-centered covolume method for elliptic problems. Numer. Math. 90 (2002) 459–486. CrossRef
Chou, S.H. and Error, Q. Li 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. CrossRef
Chou, S.H., Kwak, D.Y. and Li, Q., Lp error estimates and superconvergence for covolume or finite volume element methods. Numer. Meth. PDEs 19 (2003) 463–486. CrossRef
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978).
Dawson, C.N., Wheeler, M.F. and Woodward, C.S., A two-grid finite difference scheme for nonlinear parabolic equations. SIAM J. Numer. Anal. 35 (1998) 435–452. CrossRef
de Frutos, J. and Novo, J., Postprocessing the linear finite element method. SIAM J. Numer. Anal. 40 (2002) 805–819. CrossRef
Ewing, R.E., Lin, T. and Lin, Y., On the accuracy of the finite volume element method based on piecewise linear polynomials. SIAM J. Numer. Anal. 39 (2002) 1865–1888. CrossRef
R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods: Handbook of Numerical Analysis. North-Holland, Amsterdam (2000).
Feistauer, M., Felcman, J., Lukáčová-Medvidová, M. and Warnecke, G., Error estimates of a combined finite volume-finite element method for nonlinear convection-diffusion problems. SIAM J. Numer. Anal. 36 (1999) 1528–1548. CrossRef
García-Archilla, B., Novo, J. and Titi, E.S., Postprocessing the Galerkin method: a novel approach to approximate inertial manifolds. SIAM J. Numer. Anal. 35 (1998) 941–972. CrossRef
García-Archilla, B. and Titi, E.S., Postprocessing the Galerkin method: the finite element case. SIAM J. Numer. Anal. 37 (2000) 470–499. CrossRef
D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics 840. Springer-Verlag, New York (1989).
Lasis, A. and Süli, E., hp-version discontinuous Galerkin finite element method for semilinear parabolic problems. SIAM J. Numer. Anal. 45 (2007) 1544–1569. CrossRef
R. Li, Z. Chen and W. Wu, Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods. Marcel Dekker, New York (2000).
Ma, X., Shu, S. and Zhou, A., Symmetric finite volume discretizations for parabolic problems. Comput. Methods Appl. Mech. Engrg. 192 (2003) 4467–4485. CrossRef
Marion, M. and Error, J.C. Xu estimates on a new nonlinear Galerkin method based on two-grid finite elements. SIAM J. Numer. Anal. 32 (1995) 1170–1184. CrossRef
Rui, H., Symmetric modified finite volume element methods for self-adjoint elliptic and parabolic problems. J. Comput. Appl. Math. 146 (2002) 373–386. CrossRef
Schatz, A.H., Thomée, V. and Wahlbin, L., Maximum norm stability and error estimates in parabolic finite element equations. Comm. Pure Appl. Math. 33 (1980) 265–304. CrossRef
Sinha, R.K. and Geiser, J., Error estimates for finite volume element methods for convection-diffusion-reaction equations. Appl. Numer. Math. 57 (2007) 59–72. CrossRef
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences 68. Springer-Verlag, Berlin (1988).
V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, Berlin (1997).
Thomée, V. and Wahlbin, L., Galerkin, On methods in semilinear parabolic problems. SIAM J. Numer. Anal. 12 (1975) 378–389. CrossRef
Yan, Y., Postprocessing the finite element method for semilinear parabolic problems. SIAM J. Numer. Anal. 44 (2006) 1681–1702. CrossRef
Yang, M., A second-order finite volume element method on quadrilateral meshes for elliptic equations. ESAIM: M2AN 40 (2006) 1053–1067. CrossRef
Ye, X., A discontinuous finite volume method for the Stokes problems. SIAM J. Numer. Anal. 44 (2006) 183–198. CrossRef
Zhang, S., On domain decomposition algorithms for covolume methods for elliptic problems. Comput. Methods Appl. Mech. Engrg. 196 (2006) 24–32. CrossRef