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

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 HI-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). KEY WORDS: Helmholtz equation; finite element method; error analysis c Consider the one-dimensional case. The Helmholtz equation u" + k2u = 0 has fundamental solutions of the form u = exp( propagating waves of the form u = exp(i( ikx). These stationary solutions correspond to kx -o~t ) )

Chacterization of a 20 kHz sonoreactor. Part I: analysis of mechanical effects by classical and numerical methods

“…However, it is known [33] that, for κh=constant, the errors of the finite element solutions deteriorate rapidly as the wavenumber κ increases. This nonrobust behaviour with respect to κ is known as the pollution effect [36][37][38][39][40][41][42]. The pollution effect may be reduced by increasing the order of the elements in the finite element method or using a small enough mesh for the resolution.…”

A multi-level fast multipole BEM for 3-D elastodynamics in the frequency domain

“…acoustics [50], electromagnetics [35,39] or seismology [7,22]. In contrast with domain discretization methods, artificial boundary conditions [18] are not needed for dealing with the radiation conditions, and grid dispersion cumulative effects are absent [24,51].…”

“…acoustics [50], electromagnetics [35,39] or seismology [7,22]. In contrast with domain discretization methods, artificial boundary conditions [18] are not needed for dealing with the radiation conditions, and grid dispersion cumulative effects are absent [24,51].However, in traditional boundary element (BE) implementations, the dimensional advantage with respect to domain discretization methods is offset by the fully-populated nature of the BEM coefficient matrix, with set-up and solution times rapidly increasing with the problem size N . It is thus essential to develop alternative, faster strategies that allow to still exploit the known advantages of BEMs when large N prohibit the use of traditional implementations.…”

A simple numerical absorbing layer method in elastodynamics

“…The length of the elastic layer (left) is 4 and that of the homogeneously damped layer (right) is  (=c/f R wavelength of the longitudinal wave and c wave velocity). The size of the finite elements is /20 to have low numerical dispersion [11,12]. In the elastic medium, the element damping matrices are zero whereas homogeneous Rayleigh damping is considered in the absorbing layer by choosing identical Rayleigh coefficients for each element damping matrix in this area (the elastic properties being identical in both domains).…”

“…Such methods as finite or spectral elements have strong advantages (for complex geometries [9], nonlinear media [10], etc) but may have such drawbacks as numerical dispersion [4,11,12] for low order finite elements, or spurious reflections at the mesh boundaries [4,13]. The problem of spurious reflections may be dealt with using the Boundary Element Method [5,6,7] or coupling it with other numerical methods [14].…”