Conditioning of a Hybrid High-Order scheme on meshes with small faces

TL;DR

This paper analyzes the condition number of a Hybrid High-Order scheme for the Poisson problem, showing independence from mesh face count and robustness on cut meshes, with improvements via element aggregation.

Contribution

It provides a condition number analysis for HHO schemes on complex meshes, demonstrating stability and proposing aggregation techniques for ill-conditioned elements.

Findings

Condition number is independent of face count and element size.

HHO schemes are less affected by small-cut elements than conforming methods.

Aggregation of ill-conditioned elements improves system stability.

Abstract

We conduct a condition number analysis of a Hybrid High-Order (HHO) scheme for the Poisson problem. We find the condition number of the statically condensed system to be independent of the number of faces in each element, or the relative size between an element and its faces. The dependence of the condition number on the polynomial degree is tracked. Next, we consider HHO schemes on cut background meshes, which are commonly used in unfitted discretisations. It is well known that the linear systems obtained on these meshes can be arbitrarily ill-conditioned due to the presence of sliver-cut and small-cut elements. We show that the condition number arising from HHO schemes on such meshes is not as negatively effected as those arising from conforming methods. We describe how the condition number can be improved by aggregating ill-conditioned elements with their neighbours.