Cell Centered Finite Volume Methods Using Taylor Series Expansion Scheme Without Fictitious Domains

Abstract

The goal of this article is to study the stability and the convergence of cell-centered finite volumes (FV) in a domain $\Omega= (0,1)\times(0,1)\subset R^2$ with non-uniform rectangular control volumes. The discrete FV derivatives are obtained using the Taylor Series Expansion Scheme (TSES), (see [4] and [10]), which is valid for any quadrilateral mesh. Instead of using compactness arguments, the convergence of the FV method is obtained by comparing the FV method to the associated finite differences (FD) scheme. As an application, using the FV discretizations, convergence results are proved for elliptic equations with Dirichlet boundary condition.