Year: 2024
Author: Yunqing Huang, Shangyou Zhang
CSIAM Transactions on Applied Mathematics, Vol. 5 (2024), Iss. 2 : pp. 221–233
Abstract
In the partition of unity finite element method, the nodal basis of the standard linear Lagrange finite element is multiplied by the $P_k$ polynomial basis to form a local basis of an extended finite element space. Such a space contains the $P_1$ Lagrange element space, but is a proper subspace of the $P_{k+1}$ Lagrange element space on triangular or tetrahedral grids. It is believed that the approximation order of this extended finite element is $k,$ in $H^1$-norm, as it was proved in the first paper on the partition of unity, by Babuska and Melenk. In this work we show surprisingly the approximation order is $k+1$ in $H^1$-norm. In addition, we extend the method to rectangular/cuboid grids and give a proof to this sharp convergence order. Numerical verification is done with various partition of unity finite elements, on triangular, tetrahedral, and quadrilateral grids.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/csiam-am.SO-2023-0022
CSIAM Transactions on Applied Mathematics, Vol. 5 (2024), Iss. 2 : pp. 221–233
Published online: 2024-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 13
Keywords: Finite element partition of unity triangular grid tetrahedral grid rectangular grid.