2𝑘 superconvergence of 𝑄_{𝑘} finite elements by anisotropic mesh approximation in weighted Sobolev spaces

He, Wenming; Zhang, Zhimin

doi:10.1090/mcom/3159

$2k$ superconvergence of $Q_k$ finite elements by anisotropic mesh approximation in weighted Sobolev spaces
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by Wenming He and Zhimin Zhang;

Math. Comp. 86 (2017), 1693-1718

DOI: https://doi.org/10.1090/mcom/3159

Published electronically: October 7, 2016

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Abstract:

We first establish anisotropic finite element estimates for the discrete Green’s function of the Poisson equation. Then we prove that the bi-$k$ rectangular finite element method under a specially designed partition has the highest possible convergent rate at all element vertices for the Poisson equation on the unit square. Furthermore, we provide numerical comparison with the standard uniform rectangular partition and demonstrate that our a priori superconvergent error estimate is sharp up to a logarithm factor.

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