Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems

TL;DR

This paper develops new a posteriori error estimates for fully discrete linear parabolic problems using an adapted elliptic reconstruction technique, providing optimal error bounds in various norms.

Contribution

It introduces a novel adaptation of the elliptic reconstruction method for fully discrete schemes with changing finite element spaces, yielding optimal a posteriori error estimates.

Findings

Derived optimal a posteriori error estimates in maximum-in-time norm

Established estimates in mean-square-in-space and space-time norms

Applicable to fully discrete schemes with time-varying finite element spaces

Abstract

We derive aposteriori error estimates for fully discrete approximations to solutions of linear parabolic equations on the space-time domain. The space discretization uses finite element spaces, that are allowed to change in time. Our main tool is an appropriate adaptation of the elliptic reconstruction technique, introduced by Makridakis and Nochetto (2003). We derive novel optimal order aposteriori error estimates for the maximum-in-time and mean-square-in-space norm and the mean-square in space-time of the time-derivative norm.