(untitled) (CSIAM Transactions on Applied Mathematics, 2024)
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.