Robustness and dispersion analysis of the Partition of Unity Finite Element Method applied to the Helmholtz equation

The Partition of Unity Finite Element Method (PUFEM) has been widely used for the numerical simulation of the Helmholtz equation in different physical settings. In fact, it is a numerical pollution-free alternative method to the classical piecewise polynomial-based finite element methods. Taking into account a plane wave enrichment of the piecewise linear finite element method, the main goal of this work is focused on the derivation of the numerical dispersion relation and the robustness analysis of the PUFEM discretization when a spurious perturbation is presented in the wave number value used in the enrichment definition. From the one-dimensional Helmholtz equation, the discrete wave number is estimated based on a Bloch’s wave analysis and a priori error estimates are computed explicitly in terms of the mesh size, the wave number, and the perturbation value.