Optimal \(L_{2}\) estimates for semidiscrete Galerkin method applied to parabolic integro-differential equations with nonsmooth data

Abstract

We propose and analyse an alternate approach to a priori error estimates for the semidiscrete Galerkin approximation to a time-dependent parabolic integro-differential equation with nonsmooth initial data. The method is based on energy arguments combined with repeated use of time integration, but without using parabolic-type duality techniques. An optimal \(L_{2}\)-error estimate is derived for the semidiscrete approximation when the initial data is in \(L_{2}\). A superconvergence result is obtained and then used to prove a maximum norm estimate for parabolic integro-differential equations defined on a two-dimensional bounded domain. doi:10.1017/S1446181114000030