Conventional Finite Volumes for Flow Problems in Porous Media Involving Discontinuous Permeability: Convergence Rate Analysis of Cellwise-constant and Linear-spline Solutions

Abstract

This work develops mathematical aspects of Conventional Finite Volume schemes for flow problems in porous media governed by discontinuous absolute permeability. Focusing on incompressible one-phase flow problems in heterogeneous porous media, a particular attention is put on the homogenized absolute permeability involved in the discrete Darcy velocity over the “interaction zone” between two adjacent control volumes. The first key-step of our presentation consists in putting in place a discrete-function-space frame-work endowed with inner products and their associated norms. Then after adequate mathematical tools are deployed as projection and interpolation operators with their fundamental properties. A discrete version of the Poincaré-Friedrichs inequality is also established and used to get equivalent discrete norms. Interpolation Operators are used to define cellwise-constant and linear-spline approximate solutions. A discrete variational formulation of the finite volume problem is stated and the Lax-Milgram theorem applies (upon projection operator continuity) to show the well posedness of the discrete variational problem. A first order convergence in <i>L</i><sup>2</sup>-norm and in some discrete energy norm has been shown. Sufficient conditions to get higher order convergence rate in <i>L</i><sup>2</sup>-norm and in <i>H</i><sub>0</sub><sup>1</sup>-norm have been stated for linear-spline solutions.