Gradient recovery in adaptive finite element methods for parabolic problems

TL;DR

This paper develops rigorous energy-norm a posteriori error bounds using gradient recovery estimators for fully discrete schemes solving the heat equation, with extensive numerical validation and an adaptive method implementation.

Contribution

It provides the first rigorous derivation of ZZ estimators for fully discrete evolution problems without timestep restrictions, using elliptic reconstruction techniques.

Findings

Estimator accurately controls spatial error in heat equation schemes

Numerical experiments confirm estimator's sharpness and asymptotic behavior

Adaptive method based on estimators improves computational efficiency

Abstract

We derive energy-norm aposteriori error bounds, using gradient recovery (ZZ) estimators to control the spatial error, for fully discrete schemes for the linear heat equation. This appears to be the first completely rigorous derivation of ZZ estimators for fully discrete schemes for evolution problems, without any restrictive assumption on the timestep size. An essential tool for the analysis is the elliptic reconstruction technique. Our theoretical results are backed with extensive numerical experimentation aimed at (a) testing the practical sharpness and asymptotic behaviour of the error estimator against the error, and (b) deriving an adaptive method based on our estimators. An extra novelty provided is an implementation of a coarsening error "preindicator", with a complete implementation guide in ALBERTA.