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Monotone finite volume schemes for diffusion equations on polygonal meshes. (English) Zbl 1147.65069
Summary: We construct a nonlinear finite volume (FV) scheme for diffusion equation on star-shaped polygonal meshes and prove that the scheme is monotone, i.e., it preserves positivity of analytical solutions for strongly anisotropic and heterogeneous full tensor coefficients. Our scheme has only cell-centered unknowns, and it treats material discontinuities rigorously and offers an explicit expression for the normal flux. Numerical results are presented to show how our scheme works for preserving positivity on various distorted meshes for both smooth and non-smooth highly anisotropic solutions. The numerical results show that our scheme appears to be approximate second-order accuracy for the solution and first-order accuracy for the flux.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
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