A posteriori error estimates in the maximum norm for parabolic problems

TL;DR

This paper develops new a posteriori error estimates in the maximum norm for linear parabolic equations, using elliptic reconstruction and heat kernel estimates to improve error analysis for finite element methods.

Contribution

It introduces a simplified approach combining elliptic reconstruction with heat kernel estimates for maximum norm error bounds in parabolic problems.

Findings

A posteriori bounds in maximum norm for semidiscrete finite element approximations.

A posteriori bounds for fully discrete backward Euler finite element methods.

The elliptic reconstruction technique simplifies maximum norm error analysis.

Abstract

We derive a posteriori error estimates in the $L_\infty((0,T];L_\infty(\Omega))$ norm for approximations of solutions to linear para bolic equations. Using the elliptic reconstruction technique introduced by Makridakis and Nochetto and heat kernel estimates for linear parabolic pr oblems, we first prove a posteriori bounds in the maximum norm for semidiscrete finite element approximations. We then establish a posteriori bounds for a fully discrete backward Euler finite element approximation. The elliptic reconstruction technique greatly simplifies our development by allow\ ing the straightforward combination of heat kernel estimates with existing elliptic maximum norm error estimators.