Pointwise error analysis of the linear finite element approximation for $$-\,\Delta u + u = f$$ - Δ u + u = f in $$\Omega $$ Ω , $$\partial _n u = \tau $$ ∂ n u = τ on $$\partial \Omega $$ ∂ Ω , where $$\Omega $$ Ω is a bounded smooth domain in $$\mathbb {R}^N$$ R N , is presented. We establish $$O(h^2|\log h|)$$ O ( h 2 | log h | ) and O ( h ) error bounds in the $$L^\infty $$ L ∞ - and $$W^{1,\infty }$$ W 1 , ∞ -norms respectively, by adopting the technique of regularized Green’s functions combined with local $$H^1$$ H 1 - and $$L^2$$ L 2 -estimates in dyadic annuli. Since the computational domain $$\Omega _h$$ Ω h is only polyhedral, one has to take into account non-conformity of the approximation caused by the discrepancy $$\Omega _h \ne \Omega $$ 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.

中文翻译：

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光滑域中Neumann边值问题线性有限元法的逐点误差估计

$$-\,\Delta u + u = f$$ - Δ u + u = f in $$\Omega $$ Ω , $$\partial _n u = \tau $的线性有限元近似的逐点误差分析$ ∂ nu = τ on $$\partial \Omega $$ ∂ Ω ，其中$$\Omega $$ Ω 是$$\mathbb {R}^N$$ RN 中的有界光滑域。我们在 $$L^\infty $$ L ∞ - 和 $$W 中建立 $$O(h^2|\log h|)$$ O ( h 2 | log h | ) 和 O ( h ) 误差界限^{1,\infty }$$ W 1 , ∞ -范数，采用正则化格林函数结合局部$$H^1$$ H 1 - 和$$L^2$$ L 2 -估计在二元环中。由于计算域 $$\Omega _h$$ Ω h 只是多面体，因此必须考虑由 $$\Omega _h \ne \Omega $$ Ω h ≠ Ω 引起的近似不符合性。特别是所谓的伽辽金正交关系，在证明中使用了 3 次，并不完全成立，并且涉及域扰动项（或边界皮肤项），需要仔细处理。提供了一个数值例子来证实理论结果。