Optimal $L^2$ Error Estimates for Non-symmetric Nitsche's Methods

TL;DR

This work resolves the longstanding gap between suboptimal $L^2$ error theory and observed optimal convergence for the non-symmetric Nitsche method by crafting adjoint-consistent dual problems. It develops a sequence of regularity results and dual-argument techniques that yield $O(h^{k+1})$ convergence in the $L^2$ norm for a range of penalty regimes, including super-penalty ($\alpha\ge 1$, $c_0>0$) and penalty-free ($c_0=0$) cases, and extends the results from quasi-uniform to shape-regular meshes. The analysis hinges on tailored dual problems, refined interpolation and inf-sup arguments, and a boundary-aware Scott–Zhang approach to handle non-quasi-uniform meshes, with numerical experiments in $2$D and $3$D confirming sharpness. The findings bolster the practical use of the non-symmetric Nitsche method for boundary enforcement and interface problems, and have potential implications for unfitted finite element methods.

Abstract

We establish optimal $L^2$-error estimates for the non-symmetric Nitsche method. Existing analyses yield only suboptimal $L^2$ convergence, in contrast to consistently optimal numerical results. We resolve this discrepancy by introducing a specially constructed dual problem that restores adjoint consistency. Our analysis covers both super-penalty and penalty-free variants on quasi-uniform meshes, as well as the practically important case on general shape-regular meshes without quasi-uniformity. Numerical experiments in two and three dimensions confirm the sharpness of our theoretical results.

Figures (2)

• Figure 1: Four macro elements
• Figure 2: The first level of grid for numerical experiment

Theorems & Definitions (64)

• Lemma 1: The trace inequality
• proof
• Lemma 2: The inverse inequality
• Remark 1
• Lemma 3: MR961439
• Lemma 4
• Remark 2
• Lemma 5
• Remark 3
• Lemma 6