Node-Diamond approximation of heterogeneous and anisotropic diffusion systems on arbitrary two-dimensional grids

Abstract

We develop a new nodal numerical scheme for solving diffusion equations. Anisotropic and heterogeneous diffusion tensors are taken into account in these equations. The method allows covering a wide range of general meshes such as non-confirming and distorted ones. The main idea consists in deriving the scheme from a discrete bilinear form using cellwise approximation of the diffusion tensor and particular discrete gradients. These gradients are conceived on diamonds partitioning the cell using local geometrical objects. The degrees of freedom are placed at the centers and vertices of cells. The cell unknowns can be eliminated without any fill-in. As a result, the coercivity of the scheme holds true unconditionally by construction. The convergence theorem of the Node-Diamond scheme is proved under classical assumptions on the physical parameters of the model equation and the mesh. Numerical results show the good behavior of the proposed approach on various examples among which we consider strongly anisotropic and heterogeneous systems. For instance, optimal accuracy consisting of quadratic rates for L2-errors and linear rates for H1-errors is obtained.