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

Julien Coatléven (1)

1 IFPEN - IFP Energies nouvelles

We derive new multiple point flux approximations (MPFA) for the finite volume approximation of heterogeneous and anisotropic diffusion problems on general meshes, in dimension 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.

Domaines: Analyse numérique [math.NA]

Julien Coatléven. Unconditionally stable small stencil enriched multiple point flux approximations of heterogeneous diffusion problems on general meshes. 2022. ⟨hal-03728206v1⟩