A note on the penalty parameter in Nitsche's method for unfitted boundary value problems

TL;DR

This paper examines the choice of the stabilization parameter in Nitsche's method for unfitted boundary value problems, highlighting potential issues with error deterioration and divergence due to improper parameter selection.

Contribution

It identifies problems with common parameter setting strategies in Nitsche's method and illustrates how they can lead to increased errors or divergence in solutions.

Findings

Incorrect parameter choice can cause error deterioration.

Examples show divergence in discretization errors.

Potential pitfalls in standard stabilization strategies.

Abstract

Nitsche's method is a popular approach to implement Dirichlet-type boundary conditions in situations where a strong imposition is either inconvenient or simply not feasible. The method is widely applied in the context of unfitted finite element methods. From the classical (symmetric) Nitsche's method it is well-known that the stabilization parameter in the method has to be chosen sufficiently large to obtain unique solvability of discrete systems. In this short note we discuss an often used strategy to set the stabilization parameter and describe a possible problem that can arise from this. We show that in specific situations error bounds can deteriorate and give examples of computations where Nitsche's method yields large and even diverging discretization errors.