This article presents numerical convergence results for the multi-point flux approximation (MPFA) O-method applied to the pressure equation in 2D. The discretization is made directly in physical space, and the investigated cases are simulated on structured, but generally skew grids. Skew grids need to be used to correctly represent the physics of the underlying flow problems. Special emphasis is made on cases which impose singularities in the velocity field. Such cases frequently arise in the description of subsurface flow. Analytical tools may not be applicable to fully answer the question of convergence for such cases; in particular not for the physical space discretization.

This paper establishes the convergence of a multi point flux approximation control volume method on rough quadrilateral grids. By rough grids we refer to a family of refined quadrilateral grids where the cells are not required to approach parallelograms in the asymptotic limit. In contrast to previous convergence results for these methods we consider here a version where the flux approximation is derived directly in the physical space, and not on a reference cell. As a consequence, less regular grids are allowed. However, the extra cost is that the symmetry of the method is lost.

This paper presents the relationships between some numerical methods suitable for a heterogeneous elliptic equation with application to reservoir simulation. The methods discussed are the classical mixed finite element method (MFEM), the control-volume mixed finite element method (CVMFEM), the support operators method (SOM), the enhanced cell-centered finite difference method (ECCFDM), and the multi-point flux-approximation (MPFA) controlvolume method. These methods are all locally mass conservative, and handle general irregular grids with anisotropic and heterogeneous discontinuous permeability. In addition to this, the methods have a common weak continuity in the pressure across the edges, which in some cases corresponds to Lagrange multipliers. It seems that this last property is an essential common quality for these methods.Keywords: relationships, mixed finite element method (MFEM), expanded mixed finite element method (EMFEM), enhanced cell-centered finite difference method (ECCFDM), control-volume mixed finite element method (CVMFEM), support operator method (SOM), multi-point flux-approximation (MPFA) control-volume method