Unconditionally stable small stencil enriched multiple point flux approximations of heterogeneous diffusion problems on general meshes

Abstract

We derive new multiple point flux approximations (MPFA) for the finite volume approximation of heterogeneous and anisotropic diffusion problems on general meshes, in dimensions 2 and 3. The resulting methods are unconditionally stable while preserving the small stencil typical of MPFA finite volumes. The key idea is to solve local variational problems with a well-designed stabilization term from which we deduce conservative flux instead of directly prescribing a flux formula and solving the usual flux continuity equations. The boundary conditions of our local variational problems are handled through additional cell-centered unknowns, leading to an overall scheme with the same number of unknowns than first-order discontinuous Galerkin methods. Convergence results follow from well-established frameworks, while numerical experiments illustrate the good behavior of the method.