A convergent time-space adaptive dG(s) finite element method for parabolic problems motivated by equal error distribution

TL;DR

This paper introduces a fully discrete space-time adaptive dG(s) finite element method for linear parabolic problems, ensuring convergence through a novel a posteriori analysis and an equal error distribution principle.

Contribution

It develops a new reliable and efficient a posteriori analysis for higher order dG(s) finite element discretizations in space-time adaptive methods.

Findings

Convergence guaranteed by uniform energy estimates.

Adaptive strategy based on equal error distribution in time.

Applicable under minimal regularity assumptions.

Abstract

We shall develop a fully discrete space-time adaptive method for linear parabolic problems based on new reliable and efficient a posteriori analysis for higher order dG(s) finite element discretisations. The adaptive strategy is motivated by the principle of equally distributing the a posteriori indicators in time and the convergence of the method is guaranteed by the uniform energy estimate from [KreuzerM\"ollerSchmidtSiebert:12] under minimal assumptions on the regularity of the data.