Exact and Superconvergent Solutions of the Multi-Point Flux Approximation O-method: Analysis and Numerical Tests

Abstract

In this thesis we prove the multi-point flux approximation O-method (MPFA) to yield exact potential and flux for the trigonometric potential functions u(x,y)=sin(x)sin(y) and u(x,y)=cos(x)cos(y). This is done on uniform square grids in a homogeneous medium with principal directions of the permeability aligned with the grid directions when having periodic boundary conditions. Earlier theoretical and numerical convergence articles suggests that these potential functions should only yield second order convergence. Hence, our motivation for the analysis was to gain new insight into the convergence of the method, as well as to develop theoretical proofs for what seems as decent examples for testing implementation. An extension of the result to uniform rectangular grids in an isotropic medium is also briefly discussed, before we develop a numerical overview of the exactness phenomenon for different types of boundary conditions. Lastly, an investigation of application of these results to obtain exact potential and flux using the MPFA method for general potential functions approximated by Fourier series was conducted.