Maximum-norm a posteriori error bounds for parabolic equations discretised by the extrapolated Euler method in time and FEM in space

TL;DR

This paper develops a framework for a posteriori error estimation of linear parabolic equations discretized with Richardson extrapolation in time and finite element methods in space, using elliptic reconstructions and Green's function bounds.

Contribution

It introduces a novel a posteriori error analysis framework combining Richardson extrapolation and finite element discretizations for parabolic equations.

Findings

Provides maximum-norm a posteriori error bounds.

Utilizes elliptic reconstructions and Green's function bounds.

Designs polynomial reconstructions in time from mesh point approximations.

Abstract

A class of linear parabolic equations is considered. We derive a framework for the a posteriori error analysis of time discretisations by Richardson extrapolation of arbitrary order combined with finite element discretisations in space. We use the idea of elliptic reconstructions and certain bounds for the Green's function of the parabolic operator. The crucial point in the analysis is the design of suitable polynomial reconstructions in time from approximations that are given only in the mesh points.