Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes. SUSHI: a scheme using stabilisation and hybrid interfaces
It provides a robust numerical method for solving diffusion problems with discontinuous tensors on complex meshes, which is important for computational physics and engineering applications.
The paper develops a discretisation scheme (SUSHI) for heterogeneous anisotropic diffusion problems on general non-conforming meshes, achieving accurate numerical results and proving convergence without regularity assumptions on the solution.
A discretisation 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 which 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 on several numerical examples. Mathematical convergence of the approximate solution to the continuous solution is obtained for general (possibly discontinuous) tensors, general (possibly non-conforming) meshes, and with no regularity assumption on the solution. An error estimate is then drawn under sufficient regularity assumptions on the solution.