@Article{CSIAM-AM-5-221,
author = {Huang , Yunqing and Zhang , Shangyou},
title = {On the Optimal Order Approximation of the Partition of Unity Finite Element Method},
journal = {CSIAM Transactions on Applied Mathematics},
year = {2024},
volume = {5},
number = {2},
pages = {221--233},
abstract = {<p style="text-align: justify;">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.</p>},
issn = {2708-0579},
doi = {https://doi.org/10.4208/csiam-am.SO-2023-0022},
url = {http://global-sci.org/intro/article_detail/csiam-am/23120.html}
}