Pointwise 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 −∆u + u = f in Ω, ∂nu = τ on ∂Ω, where Ω is a bounded smooth domain in R N , is presented. We establish O(h 2 | log h|) and O(h) error bounds in the L ∞ -and W 1,∞ -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 = 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.

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