TY - JOUR T1 - On the Optimal Order Approximation of the Partition of Unity Finite Element Method AU - Huang , Yunqing AU - Zhang , Shangyou JO - CSIAM Transactions on Applied Mathematics VL - 2 SP - 221 EP - 233 PY - 2024 DA - 2024/05 SN - 5 DO - http://doi.org/10.4208/csiam-am.SO-2023-0022 UR - https://global-sci.org/intro/article_detail/csiam-am/23120.html KW - Finite element, partition of unity, triangular grid, tetrahedral grid, rectangular grid. AB -

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.