Optimal $L^2$-error estimates for the nonsymmetric Nitsche method in two dimensions

Nitsche's method is a standard device for weakly imposing Dirichlet boundary conditions, but for the stabilized nonsymmetric formulation the available $L^2$-error analysis for Poisson's equation still predicts a half-order loss, whereas numerical evidence indicates optimal convergence. We prove that, for conforming $k$th-order finite elements on quasi-uniform triangulations of convex polygonal domains in two dimensions, the stabilized nonsymmetric Nitsche approximation satisfies $$\|{u-u_h}\|_{L^2(\Omega)} \le C h^{k+1}\|{u}\|_{W^{k+1,\infty}(\Omega)}.$$ The proof compares the Nitsche solution with an auxiliary conforming finite element solution with strongly imposed projected boundary data and combines three ingredients: a two-layer boundary-strip lifting, an exact boundary identity on the one-dimensional boundary mesh, and localized residual estimates. In addition, we isolate the auxiliary $W^{1,\infty}$ estimate needed in the argument and provide a revised proof based on the $L^\infty$-stability of the boundary $L^2$-projection together with a weak discrete maximum principle for discrete harmonic functions. The analysis is intrinsically two-dimensional and clarifies why the stronger assumption $u\in W^{k+1,\infty}(\Omega)$ enters the proof.