Sharp and unified $L^2$ error estimates for the nonsymmetric Nitsche method on convex polytopes

Abstract

Nitsche's method weakly imposes Dirichlet boundary conditions, but its nonsymmetric variant has long shown a gap between theory and computation: the classical $L^2$~analysis under $H^{k+1}$~regularity predicts a half-order convergence loss, whereas numerical experiments on smooth test problems consistently produce the optimal rate. Whether this discrepancy reflects a limitation of the analysis or an essential feature of the method has remained an open question. On bounded convex polytopes in two and three dimensions, we prove a unified, regularity-dependent $L^2$~error estimate valid across the entire penalty scale $h^{-\alpha}$: \begin{align*} \|u-u_h\|_{L^2(\Omega)}\le C h^r |u|_{W^{k+1,p}(\Omega)},\qquad r=\min\bigl\{k+1,\,k+\max\{1,\alpha\}-1/p\bigr\}. \end{align*} Numerical experiments in two and three dimensions, on a one-parameter family of manufactured solutions with tunable regularity, demonstrate the sharpness of the estimate and resolve the open question. First, under merely $H^{k+1}$~regularity the half-order loss is essential; second, the optimal convergence consistently observed on smooth test problems is therefore explained by their full $W^{k+1,\infty}$~regularity, not by a limitation of the standard analysis. The theory identifies $\alpha\ge 1+1/p$ or $p=\infty$ as the sharp threshold for recovering the optimal rate $h^{k+1}$, and the experiments confirm this if-and-only-if condition in both dimensions.