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Convergence rate of a finite volume scheme for the linear convection-diffusion equation on locally refined meshes

Published online by Cambridge University Press:  15 April 2002

Yves Coudière  and
Philippe Villedieu
Yves Coudière
Affiliation:
Mathématiques pour l'Industrie et la Physique, UMR 5640, INSA, 135 avenue de Rangueil, 31077 Toulouse Cedex 4, France. (Yves.Coudiere@sophia.inria.fr)
Philippe Villedieu
Affiliation:
ONERA, Centre de Toulouse, 2 avenue Ed. Belin, 31055 Toulouse Cedex 4, France. (Philippe.Villedieu@cert.fr)
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Abstract

We study a finite volume method, used to approximate the solution of the linear two dimensional convection diffusion equation, with mixed Dirichlet and Neumann boundary conditions, on Cartesian meshes refined by an automatic technique (which leads to meshes with hanging nodes). We propose an analysis through a discrete variational approach, in a discrete H 1 finite volume space. We actually prove the convergence of the scheme in a discrete H 1 norm, with an error estimate of order O(h) (on meshes of size h).

Keywords

Finite volumesmesh refinementconvection-diffusionconvergence rate.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 6 , November 2000 , pp. 1123 - 1149
Copyright
© EDP Sciences, SMAI, 2000

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References

Bank, R.E. and Rose, D.J., Some error estimates for the box method. SIAM J. Numer. Anal. 24 (1987) 777-787. CrossRef
Baranger, J., Maitre, J.F. and Oudin, F., Connection between finite volume and mixed finite element methods. RAIRO Modél. Math. Anal. Numér. 30 (1996) 445-465. CrossRef
Berger, M.J. and Collela, P., Local adaptative mesh refinement for shock hydrodynamics. J. Comput. Phys. 82 (1989) 64-84. CrossRef
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
Cai, Z. and McCormick, S., On the accuracy of the finite volume element method for diffusion equations on composite grids. SIAM J. Numer. Anal. 27 (1990) 636-655. CrossRef
W.J. Coirier, An Adaptatively-Refined, Cartesian, Cell-based Scheme for the Euler and Navier-Stokes Equations. Ph.D. thesis, Michigan Univ., NASA Lewis Research Center (1994).
W.J. Coirier and K.G. Powell, A Cartesian, cell-based approach for adaptative-refined solutions of the Euler and Navier-Stokes equations. AIAA (1995).
Y. Coudière, Analyse de schémas volumes finis sur maillages non structurés pour des problèmes linéaires hyperboliques et elliptiques. Ph.D. thesis, Université Paul Sabatier (1999).
Y. Coudière, T. Gallouët and R. Herbin, Discrete sobolev inequalities and l p error estimates for approximate finite volume solutions of convection diffusion equation. Preprint of LATP, University of Marseille 1, 98-13 (1998).
Coudière, Y., Vila, J.P. and Villedieu, P., Convergence rate of a finite volume scheme for a two dimensionnal diffusion convection problem. ESAIM: M2AN 33 (1999) 493-516. CrossRef
Courbet, B. and Croisille, J.P., Finite volume box schemes on triangular meshes. RAIRO Modél. Math. Anal. Numér. 32 (1998) 631-649. CrossRef
M. Dauge, Elliptic Boundary Value Problems in Corner Domains. Lect. Notes Math., Springer-Verlag, Berlin (1988).
Ewing, R.E., Lazarov, R.D. and Vassilevski, P.S., Local refinement techniques for elliptic problems on cell-centered grids. I. Error analysis. Math. Comp. 56 (1991) 437-461.
R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions Eds. (to appear). Prépublication No 97-19 du LATP, UMR 6632, Marseille (1997).
Forsyth, P.A. and Sammon, P.H., Quadratic convergence for cell-centered grids. Appl. Numer. Math. 4 (1988) 377-394. CrossRef
B. Heinrich, Finite Difference Methods on Irregular Networks. Internat. Ser. Numer. Anal. 82, Birkhaüser, Verlag Basel (1987).
Herbin, R., An error estimate for a finite volume scheme for a diffusion-convection problem on a triangular mesh. Numer. Methods Partial Differential Equations 11 (1994) 165-173. CrossRef
F. Jacon and D. Knight, A Navier-Stokes algorithm for turbulent flows using an unstructured grid and flux difference splitting. AIAA (1994).
Jianguo, H. and Shitong, X., On the finite volume element method for general self-adjoint elliptic problem. SIAM J. Numer. Anal. 35 (1998) 1762-1774. CrossRef
P. Lesaint, Sur la résolution des systèmes hyperboliques du premier ordre par des méthodes d'éléments finis. Technical report, CEA (1976).
Manteuffel, T.A. and White, A.B., The numerical solution of second-order boundary values problems on nonuniform meshes. Math. Comp. 47 (1986) 511-535. CrossRef
K. Mer, Variational analysis of a mixed finite element finite volume scheme on general triangulations. Technical Report 2213, INRIA, Sophia Antipolis (1994).
Mishev, I.D., Finite volume methods on voronoï meshes. Numer. Methods Partial Differential Equations 14 (1998) 193-212. 3.0.CO;2-J>CrossRef
Morton, K.W. and Süli, E., Finite volume methods and their analysis. IMA J. Numer. Anal. 11 (1991) 241-260. CrossRef
Süli, E., Convergence of finite volume schemes for Poisson's equation on nonuniform meshes. SIAM J. Numer. Anal. 28 (1991) 1419-1430. CrossRef
J.-M. Thomas and D. Trujillo. Analysis of finite volumes methods. Technical Report 95/19, CNRS, URA 1204 (1995).
J.-M. Thomas and D. Trujillo, Convergence of finite volumes methods. Technical Report 95/20, CNRS, URA 1204 (1995).
Vanselow, R. and Scheffler, H.P., Convergence analysis of a finite volume method via a new nonconforming finite element method. Numer. Methods Partial Differential Equations 14 (1998) 213-231. 3.0.CO;2-R>CrossRef
Vassilevski, P.S., Petrova, S.I. and Lazarov, R.D.. Finite difference schemes on triangular cell-centered grids with local refinement. SIAM J. Sci. Stat. Comput. 13 (1992) 1287-1313. CrossRef
Weiser, A. and Wheeler, M.F., On convergence of block-centered finite differences for elliptic problems. SIAM J. Numer. Anal. 25 (1988) 351-375. CrossRef