A small stencil and extremum-preserving scheme for anisotropic diffusion problems on arbitrary 2D and 3D meshes

In this paper a nonlinear extremum-preserving scheme for the heterogeneous and anisotropic diffusion problems is proposed on general 2D and 3D meshes through a certain linearity-preserving approach. The so-called harmonic averaging points located at the interface of heterogeneity are employed to define the auxiliary unknowns. This new scheme is locally conservative, has only cell-centered unknowns and possesses a small stencil, which is five-point on the structured quadrilateral meshes and seven-point on the structured hexahedral meshes. The stability result in H1 norm is obtained under quite general assumptions. Numerical results show that our scheme is robust and extremum-preserving, and the optimal convergence rates are verified on general distorted meshes in case that the diffusion tensor is taken to be anisotropic, at times discontinuous.