On $p$-robust saturation on quadrangulations

TL;DR

This paper investigates how increasing local polynomial degrees in $hp$-adaptive finite element methods on quadrangulations can ensure error contraction, reducing the problem to saturation issues on reference squares with supporting numerical evidence.

Contribution

It reduces the $p$-robust saturation problem to a finite set of reference square problems and provides numerical evidence for their solutions.

Findings

Numerical evidence supports the proposed saturation solutions.

Reduction of the problem to reference squares simplifies analysis.

Insights into $p$-robust error contraction in finite element methods.

Abstract

For the Poisson problem in two dimensions, posed on a domain partitioned into axis-aligned rectangles with up to one hanging node per edge, we envision an efficient error reduction step in an instance-optimal $hp$-adaptive finite element method. Central to this is the problem: Which increase in local polynomial degree ensures $p$-robust contraction of the error in energy norm? We reduce this problem to a small number of saturation problems on the reference square, and provide strong numerical evidence for their solution.