A stabilized linearity-preserving scheme for the heterogeneous and anisotropic diffusion problems on polygonal meshes

In this paper a stabilized discretization scheme for the heterogeneous and anisotropic diffusion problems is proposed on general, possibly nonconforming polygonal meshes. The unknowns are the values at the cell center and the scheme relies on linearity-preserving criterion and the use of the so-called harmonic averaging points located at the interface of heterogeneity. The stability result and error estimate both in H1 norm are obtained under quite general and standard assumptions on polygonal meshes. The experiment results on a number of different meshes show that the scheme maintains optimal convergence rates in both L2 and H1 norms.