A least squares based diamond scheme for anisotropic diffusion problems on polygonal meshes

We present a new least squares based diamond scheme for anisotropic diffusion problems on polygonal meshes. This scheme introduces both cell-centered unknowns and vertex unknowns. The vertex unknowns are intermediate ones and are expressed as linear combinations with the surrounding cell-centered unknowns by a new vertex interpolation algorithm which is also derived in least squares approach. Both the new scheme and the vertex interpolation algorithm are applicable to diffusion problems with arbitrary diffusion tensors and do not depend on the location of discontinuity. Benefitting from the flexibility of least squares approach, the new scheme and vertex interpolation algorithm can also be extended to 3D cases naturally. The new scheme and vertex interpolation algorithm are linearity-preserving under given assumptions and the numerical results show that they maintain nearly optimal convergence rates for both L2 error and H1 error in general cases. More interesting is that a very robust performance of the new vertex interpolation algorithm on random meshes compared with the algorithm LPEW2 is found from the numerical tests.