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

Authors:
K. Lipnikov;D. Svyatskiy;Y. Vassilevski
Affiliations:
Mathematical Modeling and Analysis Group, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, United States;Mathematical Modeling and Analysis Group, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, United States;Institute of Numerical Mathematics, Russian Academy of Sciences, 8, Gubkina, 119333 Moscow, Russia
Venue:
Journal of Computational Physics
Year:
2009

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Abstract

We developed a new monotone finite volume method for diffusion equations. The second-order linear methods, such as the multipoint flux approximation, mixed finite element and mimetic finite difference methods, are not monotone on strongly anisotropic meshes or for diffusion problems with strongly anisotropic coefficients. The finite volume (FV) method with linear two-point flux approximation is monotone but not even first-order accurate in these cases. The developed monotone method is based on a nonlinear two-point flux approximation. It does not require any interpolation scheme and thus differs from other nonlinear finite volume methods based on a two-point flux approximation. The second-order convergence rate is verified with numerical experiments.