Optimal $L^2$-error estimates for the nonsymmetric Nitsche method in two dimensions
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| Format: | Preprint |
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2025
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| _version_ | 1866908977030234112 |
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| author | Chen, Gang Liu, Chaoran Zhang, Yangwen |
| author_facet | Chen, Gang Liu, Chaoran Zhang, Yangwen |
| contents | 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 \[ \|{u-u_h}\|_{L^2(Ω)} \le C h^{k+1}\|{u}\|_{W^{k+1,\infty}(Ω)}. \] 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 $u\in W^{k+1,\infty}(Ω)$ enters the proof. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_05597 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal $L^2$-error estimates for the nonsymmetric Nitsche method in two dimensions Chen, Gang Liu, Chaoran Zhang, Yangwen Numerical Analysis 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 \[ \|{u-u_h}\|_{L^2(Ω)} \le C h^{k+1}\|{u}\|_{W^{k+1,\infty}(Ω)}. \] 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 $u\in W^{k+1,\infty}(Ω)$ enters the proof. |
| title | Optimal $L^2$-error estimates for the nonsymmetric Nitsche method in two dimensions |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2510.05597 |