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A vertex‐centered linearity‐preserving discretization of diffusion problems on polygonal meshes
Author(s) -
Wu Jiming,
Gao Zhiming,
Dai Zihuan
Publication year - 2015
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.4178
Subject(s) - polygon mesh , discretization , mathematics , stencil , finite element method , volume mesh , vertex (graph theory) , finite volume method , quadrilateral , anisotropic diffusion , geometry , mesh generation , mathematical analysis , anisotropy , discrete mathematics , computational science , physics , graph , mechanics , thermodynamics , quantum mechanics
Summary This paper introduces a vertex‐centered linearity‐preserving finite volume scheme for the heterogeneous anisotropic diffusion equations on general polygonal meshes. The unknowns of this scheme are purely the values at the mesh vertices, and no auxiliary unknowns are utilized. The scheme is locally conservative with respect to the dual mesh, captures exactly the linear solutions, leads to a symmetric positive definite matrix, and yields a nine‐point stencil on structured quadrilateral meshes. The coercivity of the scheme is rigorously analyzed on arbitrary mesh size under some weak geometry assumptions. Also, the relation with the finite volume element method is discussed. Finally, some numerical tests show the optimal convergence rates for the discrete solution and flux on various mesh types and for various diffusion tensors. Copyright © 2015 John Wiley & Sons, Ltd.