Some new error estimates of a semidiscrete finite volume element method for a parabolic integro-differential equation with nonsmooth initial data

Abstract

A semidiscrete finite volume element (FVE) approximation to a parabolic integro-differential equation (PIDE) is analyzed in a two-dimensional convex polygonal domain. An optimal-order $L^2$-error estimate for smooth initial data and nearly the same optimal-order $L^2$-error estimate for nonsmooth initial data are obtained. More precisely, for homogeneous equations, an elementary energy technique and a duality argument are used to derive an error estimate of order $O\left(t^{-1}{h^2}\ln h\right)$ in the $L^2$-norm for positive time when the given initial function is only in $L^2$.