Energy-norm error estimates for finite element discretization of parabolic problems

TL;DR

This paper derives optimal energy-norm error estimates for finite element discretizations of parabolic problems, improving applicability to irregular problems by avoiding artificial regularity assumptions and using a novel error analysis approach.

Contribution

It introduces a new analysis method using L2-projection and careful treatment of time derivatives, enabling optimal error estimates without artificial regularity conditions.

Findings

Optimal energy-norm error estimates derived

Applicable to irregular problems with sub-optimal previous methods

Avoids regularity restrictions limiting previous analyses

Abstract

We consider the discretization of parabolic initial boundary value problems by finite element methods in space and a Runge-Kutta time stepping scheme. Order optimal a-priori error estimates are derived in an energy-norm under natural smoothness assumptions on the solution and without artificial regularity conditions on the parameters and the domain. The key steps in our analysis are the careful treatment of time derivatives in the H(-1)-norm and the the use of an L2-projection in the error splitting instead of the Ritz projector. This allows us to restore the optimality of the estimates with respect to smoothness assumptions on the solution and to avoid artificial regularity requirements for the problem, usually needed for the analysis of the Ritz projector, which limit the applicability of previous work. The wider applicability of our results is illustrated for two irregular problems, for which previous results can either not by applied or yield highly sub-optimal estimates.