On vertex reconstruction for cell-centered finite volume approximations of 2-D anisotropic diffusion problems

The accuracy of the diamond scheme is experimentally investigated for anisotropic diffusion problems in two space dimensions. This finite volume formulation is cell-centered on unstructured triangulations and the numerical method approximates the cell averages of the solution by a suitable discretization of the flux balance at cell boundaries. The key ingredient that allows the method to achieve second-order accuracy is the reconstruction of vertex values from cell averages. For this purpose, we review several techniques from the literature and propose a new variant of the reconstruction algorithm that is based on linear Least Squares. Our formulation unifies the treatment of internal and boundary vertices and includes information from boundaries as linear constraints of the Least Squares minimization process. It turns out that this formulation is well-posed on those unstructured triangulations that satisfy a general regularity condition. The performance of the finite volume method with different algorithms for vertex reconstructions is examined on three benchmark problems having full Dirichlet, Dirichlet-Robin and Dirichlet Neumann boundary conditions. Comparison of experimental results shows that an important improvement of the accuracy of the numerical solution is attained by using our Least Squares-based formulation. In particular, in the case of Dirichlet Neumann boundary conditions and strongly anisotropic diffusion the good behavior of the method relies on the absence of locking phenomena that appear when other reconstruction techniques are used.