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.

翻译：我们为非对称尼采方法建立了最优的$L^2$误差估计。现有分析仅能获得次优的$L^2$收敛结果，这与数值实验始终表现出的最优收敛性相矛盾。我们通过引入一个特殊构造的对偶问题来恢复伴随一致性，从而解决了这一矛盾。我们的分析涵盖了拟一致网格上的超罚分和无罚分变体，以及实际应用中重要的、无需拟一致性的广义形状正则网格情形。二维与三维数值实验证实了我们理论结果的精确性。