A Multipoint Flux Approximation with a Diamond Stencil and a Non-Linear Defect Correction Strategy for the Numerical Solution of Steady State Diffusion Problems in Heterogeneous and Anisotropic Media Satisfying the Discrete Maximum Principle

In the present paper, we solve the steady state diffusion equation in 3D domains by means of a cell-centered finite volume method that uses a Multipoint Flux Approximation with a Diamond Stencil and a Non-Linear defect correction strategy (MPFA-DNL) to guarantee the Discrete Maximum Principle (DMP). Our formulation is based in the fact that the flux of MPFA methods can be split into two different parts: a Two Point Flux Approximation (TPFA) component and the Cross-Diffusion Terms (CDT). In the linear MPFA-D method, this split is particularly simple since it lies at the core of the original method construction. In this context, we introduce a non-linear defect correction, aiming to mitigate, whenever necessary, the contributions from the CDT, avoiding, this way, spurious oscillations and DMP violations. Our new MPFA-DNL scheme is locally conservative and capable of dealing with arbitrary anisotropic diffusion tensors and unstructured meshes, without harming the second order convergence rates of the original MPFA-D. To appraise the accuracy and robustness of our formulation, we solve some benchmark problems found in literature. In this paper, we restrict ourselves to tetrahedral meshes, even though, in principle, there is no restriction to extend the method to other polyhedral control volumes.