Removing the stabilization parameter in fitted and unfitted symmetric Nitsche formulations

TL;DR

This paper introduces a parameter-free variant of Nitsche's method for finite element discretizations, removing the need for manual stabilization parameter selection while maintaining stability for boundary and interface conditions.

Contribution

The authors develop and analyze a new parameter-free Nitsche formulation for both fitted and unfitted finite element methods, simplifying implementation and ensuring stability.

Findings

The new formulations do not alter the sparsity pattern of the system matrices.

They maintain stability comparable to traditional Nitsche methods with large stabilization parameters.

The variants are easily implementable in existing finite element codes.

Abstract

In many situations with finite element discretizations it is desirable or necessary to impose boundary or interface conditions not as essential conditions -- i.e. through the finite element space -- but through the variational formulation. One popular way to do this is Nitsche's method. In Nitsche's method a stabilization parameter $\lambda$ has to be chosen "sufficiently large" to provide a stable formulation. Sometimes discretizations based on a Nitsche formulation are criticized because of the need to manually choose this parameter. While in the discontinuous Galerkin community variants of the Nitsche method -- known as "interior penalty" method in the DG context -- are known which do not require such a manually chosen stabilization parameter, this has not been considered for Nitsche formulations in other contexts. We introduce and analyse such a parameter-free variant for two applications of Nitsche's method. First, the classical Nitsche formulation for the imposition of boundary conditions with fitted meshes and secondly, an unfitted finite element discretizations for the imposition of interface conditions is considered. The introduced variants of corresponding Nitsche formulations do not change the sparsity pattern and can easily be implemented into existing finite element codes. The benefit of the new formulations is the removal of the Nitsche stabilization parameter $\lambda$ while keeping the stability properties of the original formulations for a "sufficiently large" stabilization parameter $\lambda$.