A least squares based diamond scheme for 3D heterogeneous and anisotropic diffusion problems on polyhedral meshes

Abstract

We propose a diamond scheme for 3D heterogeneous and anisotropic problem using piecewise linear approximation and least squares method. Since the continuity of the solution and flux is treated appropriately, the scheme and the vertex interpolation algorithm allow arbitrary diffusion tensors. In addition, we employ an adaptive weighted coefficient in the least squares problem of the vertex interpolation using a novel way to evaluate the magnitude of the diffusion anisotropy. Consequently our vertex interpolation has better performance than the least squares interpolation in most tested cases. The presented scheme and vertex interpolation algorithm do not need edge information, which leads to less topological searches and simplifies the programming compared with other edge-based ones. By means of unified calculations of the geometric quantities, this scheme is suitable for meshes with nonplanar faces. Numerical experiments show that our scheme is linearity-preserving and achieves nearly optimal accuracy for the solution on general meshes. Moreover, the new vertex interpolation algorithm has an ideal performance for heterogeneous and highly anisotropic diffusion problems which are tough for some existing algorithms.