2018/04/02 by Takahito Kashiwabara, Tomoya Kemmochi, Kashiwabara, Takahito +1
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.1804.00390
openalex publication_date 2018/04/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Pointwise error analysis of the linear finite element approximation for -Δu + u = f in Ω, ∂n u = τ on ∂Ω, where Ω is a bounded smooth domain in \mathbb RN, is presented. We establish O(h2|log h|) and O(h) error bounds in the L^∞- and W1,∞-norms respectively, by adopting the technique of regularized Green's functions combined with local H1- and L2-estimates in dyadic annuli. Since the computational domain Ωh is only polyhedral, one has to take into account non-conformity of the approximation caused by the discrepancy Ωh ≠ 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.