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A new finite volume scheme for anisotropic diffusion problems on general grids: convergence analysis.(English)Zbl 1112.65120

Summary: We introduce here a new finite volume scheme which was developed for the discretization of anisotropic diffusion problems; the originality of this scheme lies in the fact that we are able to prove its convergence under very weak assumptions on the discretization mesh.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs

Cite

References:

[1]Coudière, Y.; Gallouët, T.; Herbin, R., Discrete Sobolev inequalities and \(L_p\) error estimates for finite volume solutions of convection diffusion equations, M2AN Math. Model. Numer. Anal., 35, 4, 767-778 (2001) ·Zbl 0990.65122
[2]Coudière, Y.; Vila, J.-P.; Villedieu, Ph., Convergence rate of a finite volume scheme for a two-dimensional convection-diffusion problem, M2AN Math. Model. Numer. Anal., 33, 3, 493-516 (1999) ·Zbl 0937.65116
[3]Domelevo, K.; Omnes, P., A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids, M2AN Math. Model. Numer. Anal., 39, 6, 1203-1249 (2005) ·Zbl 1086.65108
[4]Eymard, R.; Gallouët, T., H-convergence and numerical schemes for elliptic equations, SIAM J. Numer. Anal., 41, 2, 539-562 (2000) ·Zbl 1049.35015
[5]Eymard, R.; Gallouët, T.; Herbin, R., A cell-centered finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension, IMA J. Numer. Anal., 26, 2, 326-353 (2006) ·Zbl 1093.65110
[6]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, 2, 165-173 (1995) ·Zbl 0822.65085
[7]Le Potier, C., Schéma volumes finis pour des opérateurs de diffusion fortement anisotropes sur des maillages non structurés, C. R. Math. Acad. Sci. Paris, Ser. I, 340, 12, 921-926 (2005) ·Zbl 1076.76049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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