Reliability of infinite element methods for the numerical computation of waves

Abstract The problem of time-harmonic acoustic scattering and fluid-solid interaction are formulated as variational models based on the Helmholtz equation in an exterior domain or on a system of coupled Helmholtz equations, respectively. The quality of discrete solutions depends on the relation between the physical and geometrical parameters (frequency, domain) and the numerical parameters (stepsize, order of polynomial approximation). In the coupled case, the relation of material constants fluid/solid is also of essential influence. We investigate the dependence of the FEM solution on these parameters. In particular, we address the specific behaviour for high frequency and give a survey of results on the h-version and the h-p-version of the Galerkin FEM. We show that the finite element error is generally polluted and investigate the dependence of the pollution term on the wave number k, on the direction of the wave vector k (in the 2D case) and on the material properties of different acoustic media in the coupled problem. Finally, we comment on the influence of the parameters on a posteriori estimators. We conclude that the traditional rules of mesh design and a posteriori error estimation are reliable only in the so-called asymptotic range which is not achievable in practical computations for high frequencies. Mesh design in that case should be based on the so-called pre-asymptotic estimates which take the pollution error into account. The theoretical results are illustrated by numerical evaluation of simple model problems.