An edge-centered scheme for anisotropic diffusion problems with discontinuities on distorted quadrilateral meshes

We propose a Tailored Finite Point Method for the solution of transient anisotropic diffusion problems on distorted quadrilateral meshes. We establish an edge-centered scheme based on the flux continuity. The main feature of the scheme lies in the fact that the approximate solutions as well as the gradients are determined by using a linear combination of special solutions. These special solutions are required to satisfy the homogeneous form of approximated equations on the local cell. This brings two important assets to numerical solution of the problems. First, the scheme efficiently tailors itself fit into the local properties of the solution, so that it exhibits robustness to mesh distortion and can correctly reproduce the details of the interface layers even without refining the mesh. Second, the flux can be approximated in terms of the edge-centered unknowns defined at the local cell and the unknowns at the adjacent cells are not needed. The local stencil makes writing code for the continuity condition of the flux on each edge easier. Numerical experiments are carried out to validate the performance of the scheme with or without discontinuities on various types of distorted quadrilateral meshes.