$L^\infty$- and $W^{1,\infty}$-error estimates of linear finite element method for Neumann boundary value problems in a smooth domain
Provides rigorous error bounds for finite element methods on Neumann problems with curved boundaries, addressing a known technical challenge for practitioners.
The paper establishes optimal pointwise error estimates for linear finite element approximation of Neumann problems in smooth domains, achieving $O(h^2|\\log h|)$ in $L^\\infty$ and $O(h)$ in $W^{1,\\infty}$, despite domain mismatch. Numerical results confirm the theory.
Pointwise error analysis of the linear finite element approximation for $-Δu + u = f$ in $Ω$, $\partial_n u = τ$ on $\partialΩ$, where $Ω$ is a bounded smooth domain in $\mathbb R^N$, is presented. We establish $O(h^2|\log h|)$ and $O(h)$ error bounds in the $L^\infty$- and $W^{1,\infty}$-norms respectively, by adopting the technique of regularized Green's functions combined with local $H^1$- and $L^2$-estimates in dyadic annuli. Since the computational domain $Ω_h$ is only polyhedral, one has to take into account non-conformity of the approximation caused by the discrepancy $Ω_h \neq Ω$. In particular, the so-called Galerkin orthogonality relation, utilized three times in the proof, does not exactly hold and involves domain perturbation terms (or boundary-skin terms), which need to be addressed carefully. A numerical example is provided to confirm the theoretical result.