Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation

TLDR

Numerical solutions of the Helmholtz equation produce dispersive waves whose phase velocity depends on the wavenumber, and dispersion analysis examines the phase difference between exact and numerical solutions, while recent error estimates include a pollution term. The study evaluates the authors’ recent one‑dimensional phase‑difference result through numerical experiments and situates it within the broader literature on the topic. The authors present theoretical results alongside numerical tests on several model problems, focusing on the performance of the Galerkin hp‑version with higher‑order polynomial approximations. They show that earlier H¹ error bounds are nondispersive and only valid for medium or high wavenumbers on extremely refined meshes, that the pollution term scales with the phase difference under certain assumptions, and that this establishes a connection between dispersion analysis and numerical analysis.

Abstract

Abstract When applying numerical methods for the computation of stationary waves from the Helmholtz equation, one obtains ‘numerical waves’ that are dispersive also in non‐dispersive media. The numerical wave displays a phase velocity that depends on the parameter k of the Helmholtz equation. In dispersion analysis, the phase difference between the exact and the numerical solutions is investigated. In this paper, the authors' recent result on the phase difference for one‐dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic. It is then shown that previous error estimates in H 1 ‐norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only. On the other hand, recently proven error estimates for constant resolution contain a pollution term. With certain assumptions on the exact solution, this term is of the order of the phase difference. Thus a link is established between the results of dispersion analysis and the results of numerical analysis. Throughout the paper, the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems. Special attention is given to the performance of the Galerkin method with a higher order of polynomial approximation p ( h ‐ p ‐version).