Numerical Dispersion of Local‐Domain Trigonometric Function‐Enriched Finite Elements for Wave Propagation Analysis

ABSTRACT This paper presents the numerical dispersion analysis of a recently developed local‐domain‐enriched finite element (LEFE) for wave propagation problems. LEFE enriches standard finite element interpolations with trigonometric functions defined in the element domain, which satisfy the partition of unity and vanish at the nodes. A formulation is developed to determine numerical dispersion errors from spatial and temporal discretizations, with explicit expressions derived for the characteristic equation linking numerical phase velocity to the non‐dimensional wavenumber and Courant‐Friedrichs‐Lewy (CFL) number. Based on this, a sampling theorem is established to estimate optimal element size and time step for accurate analysis. Results show that LEFE with one and two enrichment terms requires 9‐ and 18‐times fewer elements per wavelength than the conventional finite element method (FEM) and significantly outperforms the other existing enriched elements. Unlike the latter elements, LEFE exhibits a monotonic reduction in combined dispersion error with decreasing CFL number. Further, it produces a unique relation between the total‐to‐spatial dispersion ratio and the product of element size, wavenumber, and CFL number, enabling a generalized time‐step criterion. LEFE achieves 7‐18‐fold reductions in degrees of freedom (DOFs) and 24350‐fold reductions in computational time compared to standard FEM, demonstrating its high accuracy and efficiency in simulating high‐frequency narrowband and broadband wave propagation.