Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces

A symmetric discretization scheme for heterogeneous anisotropic diffusion problems on general meshes is developed and studied. The unknowns of this scheme are the values at the centre of the control volumes and at some internal interfaces that may, for instance, be chosen at the diffusion tensor discontinuities. The scheme is therefore completely cell centred if no edge unknown is kept. It is shown to be accurate for several numerical examples. The convergence of the approximate solution to the continuous solution is proved for general (possibly discontinuous) tensors and general (possibly nonconforming) meshes and with no regularity assumption on the solution. An error estimate is then deduced under suitable regularity assumptions on the solution.