$L^{\infty }$- and $W^{1,\infty }$-error estimates of linear finite element method for Neumann boundary value problems in a smooth domain

Abstract

Pointwise error analysis of the linear finite element approximation for $-\Delta u+u=f$ in $\Omega$, ${\partial }_{n}u=\tau$ on $\partial \Omega$, where $\Omega$ 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 ${\Omega }_h$ is only polyhedral, one has to take into account non-conformity of the approximation caused by the discrepancy ${\Omega }_h\ne \Omega$. 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.

21 pages

Cited by in corpus (2)

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