On the Optimal Order Approximation of the Partition of Unity Finite Element Method

On the Optimal Order Approximation of the Partition of Unity Finite Element Method

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.

Author Details

Yunqing Huang

Shangyou Zhang