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

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.