On p-Robust Saturation for hp-AFEM

TL;DR

This paper analyzes a p-robust error estimator within an hp-adaptive finite element method, focusing on how polynomial degrees should be increased for p-independent error reduction and proposing an optimal hp-adaptive approach.

Contribution

It introduces a p-robust equilibrated flux estimator and investigates degree refinement strategies to achieve p-independent error reduction, enabling an instance optimal hp-adaptive method.

Findings

The p-robust estimator effectively guides degree refinement.

A strategy for increasing polynomial degrees to ensure p-independent error reduction.

The method can be extended to an instance optimal hp-adaptive scheme with coarsening.

Abstract

We consider the standard adaptive finite element loop SOLVE, ESTIMATE, MARK, REFINE, with ESTIMATE being implemented using the $p$-robust equilibrated flux estimator, and MARK being D\"orfler marking. As a refinement strategy we employ $p$-refinement. We investigate the question by which amount the local polynomial degree on any marked patch has to be increase in order to achieve a $p$-independent error reduction. The resulting adaptive method can be turned into an instance optimal $hp$-adaptive method by the addition of a coarsening routine.