An Alternate Approach to Optimal L 2-Error Analysis of Semidiscrete Galerkin Methods for Linear Parabolic Problems with Nonsmooth Initial Data

Abstract

In this article, we propose and analyze an alternate proof of a priori error estimates for semidiscrete Galerkin approximations to a general second order linear parabolic initial and boundary value problem with rough initial data. Our analysis is based on energy arguments without using parabolic duality. Further, it follows the spirit of the proof technique used for deriving optimal error estimates for finite element approximations to parabolic problems with smooth initial data and hence, it unifies both theories, that is, one for smooth initial data and other for nonsmooth data. Moreover, the proposed technique is also extended to a semidiscrete mixed method for linear parabolic problems. In both cases, optimal L 2-error estimates are derived, when the initial data is in L 2. A superconvergence phenomenon is also observed, which is then used to prove L ∞-estimates for linear parabolic problems defined on two-dimensional spatial domain again with rough initial data.