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Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes
Author(s) -
Andreianov Boris,
Boyer Franck,
Hubert Florence
Publication year - 2007
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20170
Subject(s) - mathematics , polygon mesh , discretization , duality (order theory) , finite volume method , nonlinear system , partial differential equation , boundary value problem , dirichlet distribution , type (biology) , laplace operator , mathematical analysis , discrete mathematics , geometry , ecology , physics , quantum mechanics , mechanics , biology
Abstract Discrete duality finite volume schemes on general meshes, introduced by Hermeline and Domelevo and Omnès for the Laplace equation, are proposed for nonlinear diffusion problems in 2D with nonhomogeneous Dirichlet boundary condition. This approach allows the discretization of non linear fluxes in such a way that the discrete operator inherits the key properties of the continuous one. Furthermore, it is well adapted to very general meshes including the case of nonconformal locally refined meshes. We show that the approximate solution exists and is unique, which is not obvious since the scheme is nonlinear. We prove that, for general W −1, p ′ (Ω) source term and W 1‐(1/ p ), p (∂Ω) boundary data, the approximate solution and its discrete gradient converge strongly towards the exact solution and its gradient, respectively, in appropriate Lebesgue spaces. Finally, error estimates are given in the case where the solution is assumed to be in W 2, p ( Ω ). Numerical examples are given, including those on locally refined meshes. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007