Maximum-norm a posteriori error bounds for an extrapolated Euler/finite element discretisation of parabolic equations

TL;DR

This paper develops maximum-norm a posteriori error bounds for an extrapolated Euler finite element method applied to linear parabolic equations, using elliptic reconstructions and Green's function bounds.

Contribution

It introduces a novel a posteriori error estimation technique combining extrapolation, elliptic reconstructions, and Green's function bounds for maximum-norm error control.

Findings

Provides maximum-norm a posteriori error bounds for the method.

Demonstrates the effectiveness of the error estimates through theoretical analysis.

Extends existing techniques to include extrapolated Euler and finite element discretizations.

Abstract

A class of linear parabolic equations are considered. We give a posteriori error estimates in the maximum norm for a method that comprises extrapolation applied to the backward Euler method in time and finite element discretisations in space. We use the idea of elliptic reconstructions and certain bounds for the Green's function of the parabolic operator.