Error estimates for semidiscrete Galerkin approximation to a time dependent parabolic integro-differential equation with nonsmooth data

Abstract

In this paper, an attempt has been made to carry over known results for the finite element Galerkin method for a time dependent parabolic equation with nonsmooth initial data to an integro-differential equation of parabolic type. More precisely, for the homogeneous problem a standard energy technique in conjunction with a duality argument is used to obtain an L²-error estimate of order O (h²/t) for the semidiscrete solution when the given initial function is only in L². Further, for the nonhomogeneous case with zero initial condition, an error estimate of order O (h² log (1/h)) uniformly in time is proved, provided that the nonhomogeneous term is in L^∞(L²). The present paper provides a complete answer to an open problem posed on p. 106 of the book Finite Element Methods for Integro-differential Equations by Chen and Shih.