High-order finite element solutions for Helmholtz problems

Abstract

The approximation of wave problems, particularly at high frequencies, with the classical finite element method (FEM) requires very fine mesh grids. The requirement of a dense mesh, however, leads to approximations that are prohibitively expensive to calculate, in terms of memory storage. For this reason, high-order numerical techniques have been developed to attack the limitation of FEM. The high-order finite element formulations are examined for the solution of two-dimensional Helmholtz wave problems. Considered numerical methods in this work include, the Lagrangian polynomial finite element method with uniform nodal distribution (FEM), Lagrangian polynomial finite elements based on Chebyshev-Gauss-Lobatto nodal distribution to give the spectral element method (SEM), the partition of unity finite element method (PUFEM) and the discontinuous enrichment method (DEM). PUFEM and DEM use propagating plane wave basis functions in the finite element approximation. Enriching the solution space with plane wave functions provides a remarkable reduction of the computational cost, in comparison to polynomial based element methods but, ill-conditioning is shown to be an inherent feature when too many plane waves are used in an element. The principal objective of this work is to assessed the considered finite element formulations. They are compared in terms of accuracy and conditioning against various measures of cost. The considered test examples include wave scattering by a rigid circular cylinder, evanescent wave cases and propagation of waves in a duct with rigid walls. This work also exploits the PUFEM approach with different enrichment functions, such as Bessel or evanescent wave functions, when singular or rapidly decaying fields are present.