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.

划分有限元方法（PUFEM）已被广泛用于在不同物理环境下对亥姆霍兹方程进行数值模拟。实际上，它是经典的基于分段多项式的有限元方法的一种数值无污染的替代方法。考虑到分段线性有限元方法的平面波富集，这项工作的主要目标集中在数值色散关系的推导以及在波数值中出现杂散扰动时PUFEM离散化的鲁棒性分析。在浓缩定义中使用。从一维亥姆霍兹方程，基于布洛赫波分析和先验估计离散波数 根据网格大小，波数和摄动值显式计算出误差估计。