Error Estimates for Nitsche's Method on Approximate Domains

TL;DR

This paper provides detailed error estimates for Nitsche's method on approximate domains, highlighting how geometric errors influence convergence in unfitted finite element methods.

Contribution

It introduces refined error bounds that distinguish effects of boundary location and normal perturbations across different norms.

Findings

Energy norm amplifies boundary location errors by h^{-1/2}

Refined H^1-seminorm estimate removes this amplification

Optimal L^2-error estimate decouples geometry errors from mesh size

Abstract

We derive a priori error estimates for Nitsche's method applied to elliptic problems on approximate domains. Such approximations arise, for example, in unfitted finite element methods, data-driven simulations, and evolving domain problems, where the computational domain does not coincide exactly with the physical one. We quantify geometric errors in terms of boundary location and normal perturbations and carry out the analysis in an abstract CutFEM framework under standard stability assumptions. In the energy norm, we obtain an estimate exhibiting an $h^{-1/2}$ amplification of the boundary location error. We then prove a refined $H^1$-seminorm estimate that removes this amplification, yielding a sharper bound with additive contributions from boundary location and normal errors. Finally, we establish an optimal order $L^2$-error estimate based on a refined duality argument, where the geometry contribution appears as a separate additive term, decoupled from the mesh size $h$. The results reveal a fundamental distinction between the norms: the energy norm amplifies boundary location errors while remaining insensitive to normal perturbations, the $H^1$-seminorm separates location and normal errors, and the $L^2$-norm is insensitive to normal perturbations. This provides a clear characterization of how geometric approximation affects convergence in Nitsche-based finite element methods, with particular relevance for unfitted discretizations.