On the Optimal Order Approximation of the Partition of Unity Finite Element Method (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 forma local basis of an extended finite element space. Such a space contains the $P_1$ Lagrangeelement 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 extendedfinite element is $k,$ in $H^1$-norm, as it was proved in the first paper on the partition ofunity, by Babuska and Melenk. In this work we show surprisingly the approximationorder is $k+1$ in $H^1$-norm. In addition, we extend the method to rectangular/cuboidgrids and give a proof to this sharp convergence order. Numerical verification is donewith various partition of unity finite elements, on triangular, tetrahedral, and quadrilateral grids.
Keywords: Finite element; partition of unity; triangular grid; tetrahedral grid; rectangular grid.