On the Finite Volume Element Version of Ritz-Volterra Projection and Applications to Related Equations

Abstract

In this paper, we present a general error analysis framework for the finite volume element (FVE) approximation to the Ritz-Volterra projection, the Sobolev equations and parabolic integro-differential equations. The main idea in our paper is to consider the FVE methods as perturbations of standard finite element methods which enables us to derive the optimal $L_2$ and $H^1$ norm error estimates, and the $L_\infty$ and $W^1_\infty$ norm error estimates by means of the time dependent Green functions. Our discussions also include elliptic and parabolic problems as the special cases.