Construction and Convergence Study of Schemes Preserving the Elliptic Local Maximum Principle

Authors:
Jérôme Droniou;Christophe Le Potier
Affiliations:
droniou@math.univ-montp2.fr;clepotier@cea.fr
Venue:
SIAM Journal on Numerical Analysis
Year:
2011

Citing 12
Cited 2

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Discretization on Unstructured Grids For Inhomogeneous, Anisotropic Media. Part II: Discussion And Numerical Results

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Interpolation-free monotone finite volume method for diffusion equations on polygonal meshes

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Abstract

We present a method to approximate (in any space dimension) diffusion equations with schemes having a specific structure; this structure ensures that the discrete local maximum and minimum principles are respected, and that no spurious oscillations appear in the solutions. When applied in a transient setting on models of concentration equations, it guaranties in particular that the approximate solutions stay between the physical bounds. We make a theoretical study of the constructed schemes, proving under a coercivity assumption that their solutions converge to the solution of the PDE. Several numerical results are also provided; they help us understand how the parameters of the method should be chosen. These results also show the practical efficiency of the method, even when applied to complex models.