[Paper Review] On the Design of Locking Free Ghost Penalty Stabilization and the Relation to CutFEM with Discrete Extension

Erik Burman, Peter Hansbo+1 more|arXiv (Cornell University)|May 3, 2022

Numerical methods for differential equations4 citations

TL;DR

This paper introduces a novel ghost penalty stabilization for cut finite element methods that generalizes previous approaches by allowing stabilization of arbitrary functionals—such as individual degrees of freedom—and by enabling connections between elements beyond face neighbors. The key contribution is proving that nodal stabilization satisfies robustness conditions, enabling the stabilization parameter to tend to infinity without locking, thereby establishing a direct equivalence to discrete extension operators in CutFEM.

ABSTRACT

In this note, we develop a new stabilization mechanism for cut finite element methods that generalizes previous approaches of ghost penalty type in two ways: (1) The quantity that is stabilized and (2) The choice of elements that are connected in the stabilization. In particular, we can stabilize functionals of the discrete function such as finite element degrees of freedom. We subsequently show that the kernel of our ghost penalty operator defines a finite element space based on discrete extensions in the spirit of those introduced in Burman, E.; Hansbo, P. and Larson, M. G., CutFEM Based on Extended Finite Element Spaces, arXiv2101.10052, 2021.

Motivation & Objective

• To develop a generalized ghost penalty stabilization framework that extends beyond standard element-based stabilization in cut finite element methods.
• To enable stabilization of arbitrary functionals of the discrete solution, such as individual degrees of freedom, for more precise control of unstable modes.
• To show that the generalized stabilization avoids locking even as the penalty parameter tends to infinity, ensuring robustness.
• To establish a direct theoretical and algorithmic connection between ghost penalty stabilization and discrete extension operators in CutFEM.
• To provide a unified framework that supports optimal order convergence and condition number estimates for second-order elliptic problems.

Proposed method

• Proposes a generalized ghost penalty form where the stabilized quantity is a functional of the discrete function, including nodal degrees of freedom.
• Introduces a flexible coupling strategy allowing elements intersecting the boundary to be stabilized via elements within a distance proportional to the mesh size, not just face neighbors.
• Defines a stabilization form that acts directly on degrees of freedom, enabling strong enforcement of constraints in the limit of infinite penalty parameter.
• Uses a discrete extension operator framework to relate the ghost penalty to extension-based stabilization, showing equivalence in the limit of large stabilization parameters.
• Applies inverse inequalities and stability estimates for canonical extensions to derive bounds on the stabilization form.
• Establishes coercivity of the discrete bilinear form under sufficient conditions on the localization and strength of the penalty terms.

Experimental results

Research questions

• RQ1 Can ghost penalty stabilization be generalized to stabilize arbitrary functionals of the discrete solution, such as individual degrees of freedom, rather than just element-based quantities?
• RQ2 Does a generalized ghost penalty formulation remain robust and locking-free when the stabilization parameter tends to infinity?
• RQ3 How does the proposed nodal stabilization relate to the discrete extension operator framework in CutFEM?
• RQ4 What conditions on the coupling between elements ensure stability and optimal convergence in cut finite element methods?
• RQ5 Can the generalized ghost penalty form be shown to satisfy the abstract framework for stability and optimal error estimates?

Key findings

• The proposed nodal stabilization satisfies the necessary conditions for robustness and optimal convergence, even as the stabilization parameter tends to infinity.
• The kernel of the ghost penalty operator corresponds exactly to the finite element space defined by discrete extensions, establishing a direct equivalence between stabilization and extension approaches.
• The generalized ghost penalty form ensures optimal order a priori error estimates and condition number bounds for second-order elliptic problems.
• The method achieves stability and optimal convergence without locking, even in the limit of infinite penalty parameters, by satisfying the required coercivity and consistency conditions.
• The theoretical framework extends naturally to higher-order elliptic problems, as shown by comparison with companion works.
• The analysis confirms that the stabilization form is coercive under sufficient localization of couplings and appropriate choice of penalty parameters.