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Error Estimates for Nitsche's Method on Approximate Domains

Mats G. Larson, Karl Larsson, Shantiram Mahata

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.

Error Estimates for Nitsche's Method on Approximate Domains

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 amplification of the boundary location error. We then prove a refined -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 -error estimate based on a refined duality argument, where the geometry contribution appears as a separate additive term, decoupled from the mesh size . The results reveal a fundamental distinction between the norms: the energy norm amplifies boundary location errors while remaining insensitive to normal perturbations, the -seminorm separates location and normal errors, and the -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.

Paper Structure

This paper contains 19 sections, 109 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Outline.
  4. The Problem and Method
  5. The Problem.
  6. Approximation Space.
  7. CutFEM on the Approximate Domain.
  8. Stabilization Form.
  9. Properties of the Method.
  10. The Standard Error Estimates
  11. Interpolation Estimate.
  12. Refined Error Estimates
  13. Numerical Examples
  14. Summary of Geometric Requirements.
  15. Implementation and Parameter Choices.
  16. ...and 4 more sections

Figures (6)

  • Figure 1: $\delta$-study mesh and solutions. Illustration of the mesh and numerical solution for two mesh sizes, $h_0$ and $h_1=h_0/2$, when the boundary perturbation is chosen as $\delta = h^{p+1/2}$. The scaling of $\delta$ with $h$ results in different perturbed domains on each mesh.
  • Figure 2: $\delta$-scaling. Convergence in the energy norm, $H^1$-seminorm, and $L^2$-norm for polynomial orders $p=1,2,3$ and different scalings $\delta = h^\alpha$. The results illustrate how the choice of $\alpha$ affects the convergence rate, and confirm that optimal order convergence is obtained when $\delta$ scales according to the theoretical predictions for each norm. The slight loss of convergence rate observed for $p=3$ in the energy and $L^2$ norms is attributed to numerical inaccuracies in the evaluation of the series expansion used to compute the reference solution.
  • Figure 3: Normal study mesh and solutions. Illustration of the mesh and numerical solution for two mesh sizes, $h_0$ and $h_1=h_0/2$, for different values of the parameter $\alpha_n$ in \ref{['eq:normal-perturbation']}. The boundary perturbation is scaled as $\delta = h^p$, while $\alpha_n$ controls the oscillation frequency and thereby the accuracy of the normal approximation, yielding $\delta_n \sim h^{-\alpha_n}\delta$.
  • Figure 4: Normal study. Convergence in the energy norm, $H^1$-seminorm, and $L^2$-norm for $p=2$ and varying values of $\alpha_n$. The energy norm and $L^2$-norm retain optimal convergence for all $\alpha_n$, while the $H^1$-seminorm deteriorates for $\alpha_n > 0$, in agreement with the analysis. The $L^2$-errors decrease slightly with increasing $\alpha_n$, suggesting a mild stabilizing effect of the higher oscillation frequency.
  • Figure 5: Level-set mesh and solutions. Illustration of the mesh and numerical solution for two mesh sizes, $h_0$ and $h_1=h_0/2$, when a piecewise linear level-set function is used to describe the domain. The resulting polygonal boundary provides a geometric approximation with $\delta \sim h^2$ and $\delta_n \sim h$.
  • ...and 1 more figures

Theorems & Definitions (5)

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