Convergence of a Cell-Centered Finite Volume Method and Application to Elliptic Equations

Abstract

We study the consistency and convergence of the cell-centered Finite Volume (FV) external approximation of $H^1_0(\Omega)$, where a 2D polygonal domain $\Omega$ is discretized by a mesh of convex quadrilaterals. The discrete FV derivatives are defined by using the so-called Taylor Series Expansion Scheme (TSES). By introducing the Finite Difference (FD) space associated with the FV space, and comparing the FV and FD spaces, we prove the convergence of the FV external approximation by using the consistency and convergence of the FD method. As an application, we construct the discrete FV approximation of some typical elliptic equations, and show the convergence of the discrete FV approximations to the exact solutions.