Linking ghost penalty and aggregated unfitted methods

TL;DR

This paper investigates the relationship between ghost penalty stabilization and aggregation-based finite element methods for elliptic PDEs on unfitted meshes, proposing new locking-free ghost penalty methods that converge to aggregated finite element schemes.

Contribution

It introduces novel ghost penalty methods that are locking-free and connect to aggregation-based finite element methods, with comprehensive numerical validation.

Findings

Ghost penalty methods suffer locking as penalty parameter increases.

Aggregated finite element spaces are locking-free due to their extension operator structure.

Proposed methods are locking-free and converge to aggregated finite element methods.

Abstract

In this work, we analyse the links between ghost penalty stabilisation and aggregation-based discrete extension operators for the numerical approximation of elliptic partial differential equations on unfitted meshes. We explore the behavior of ghost penalty methods in the limit as the penalty parameter goes to infinity, which returns a strong version of these methods. We observe that these methods suffer locking in that limit. On the contrary, aggregated finite element spaces are locking-free because they can be expressed as an extension operator from well-posed to ill-posed degrees of freedom. Next, we propose novel ghost penalty methods that penalise the distance between the solution and its aggregation-based discrete extension. These methods are locking-free and converge to aggregated finite element methods in the infinite penalty parameter limit. We include an exhaustive set of numerical experiments in which we compare weak (ghost penalty) and strong (aggregated finite elements) schemes in terms of error quantities, condition numbers and sensitivity with respect to penalty coefficients on different geometries, intersection locations and mesh topologies.