Error estimates for semidiscrete finite element methods for parabolic integro-differential equations

Abstract

The purpose of this paper is to attempt to carry over known results for spatially discrete finite element methods for linear parabolic equations to integro-differential equations of parabolic type with an integral kernel consisting of a partial differential operator of order β ≤ 2 \beta \leq 2 . It is shown first that this is possible without restrictions when the exact solution is smooth. In the case of a homogeneous equation with nonsmooth initial data v , v ∈ L 2 v,v \in {L_2} , optimal O ( h r ) O({h^r}) convergence for positive time is possible in general only if r ≤ 4 − β r \leq 4 - \beta . This depends on the fact that the exact solution is then only in H 4 − β {H^{4 - \beta }} .