A simplified enhanced MPFA formulation for the elliptic equation on general grids

Abstract

Multi-point flux approximation (MPFA) has proven to be a powerful tool for discretizing the diffusion equation on general grids with heterogeneous anisotropic permeability tensors and hence, removing the O(1) error introduced by two-point flux approximation (TPFA) for non-K-orthogonal grids. However, it is well known that the classical MPFA-O suffers from monotonicity issues and strong unphysical oscillations can be present for highly anisotropic media. Enriched MPFA (EMPFA) and MPFA with full pressure support (FPS) have been proposed in the literature to reduce the strength of the oscillations. In this work, we present a simplified enhanced MPFA formulation (eMPFA) on general grids in 2D. Similar to the MPFA-O method, our formulation starts with an interaction region formed around the vertex of control volumes. Potential at the vertex is introduced as an auxiliary unknown in addition to potential at the centroids of the control volume faces. The original EMPFA and FPS solve the diffusion equation on a small volume around the vertex to close the local system of equations. To simplify the formulation, we propose an average technique to approximate potential at the vertex. Utilizing the potential value at the vertex, full potential continuity can be imposed on control volume faces to construct a more accurate flux across control volume faces. Extensive numerical experiments are conducted to test the eMPFA formulation. Specifically, we compare eMPFA with MPFA-O and two variants of EMPFA in detail. The results show that our average technique is quite robust even for highly distorted quadrilateral grids with large permeability anisotropy ratios and solutions of our formulation are in excellent agreement with EMPFA using bilinear pressure support. The new scheme also reproduces linear potential solutions and has comparable, and in some cases even better, convergence properties to those of the MPFA-O method. Finally, we generalize the formulation to include structured and unstructured triangular and polygonal grids and present the results.