Analysis of a New MPFA Formulation for Flow Problems in Geologically Complex Media

This work analyzes some mathematical aspects of a new Multi-Point Flux Approximation (MPFA) formulation for flow problems. This MPFA formulation has been developed in [12, 13] for quadrilateral grids and [10] for unstructured grids. Our MPFA formulation displays capabilities for handling flow problems in geologically complex media modelled by spatially varying full permeability tensor. However in this work, we focus our attention on the case of anisotropic homogeneous porous media. In this framework, the proposed MPFA formulation leads to a well-posed discrete problem which is a linear system whose associated matrix is symmetric and positive definite, even if the permeability tensor governing the flow is only positive definite. Following the spirit of the finite element theory, we have introduced the concept of globally continuous and piecewise linear approximate solution. The convergence analysis of this solution is strongly based upon another concept: the weak approximate solution. Stability and convergence results for the weak approximate solution are proven for L2- and L∞-norm, and for a discrete energy norm as well. These results permit to prove some error estimates related to the globally continuous and piecewise linear approximate solution.