Spectral Approximation of the Helmholtz Equation with High Wave Numbers

Abstract

Abstract.A complete error analysis is performed for the spectral-Galerkin approximation of a model Helmholtz equation with high wave numbers. The analysis presented in this paper does not rely on the explicit knowledge of continuous/discrete Green's functions and does not require any mesh condition to be satisfied. Furthermore, new error estimates are also established for multidimensional radial and spherical symmetric domains. Illustrative numerical results in agreement with the theoretical analysis are presented.

Analysis of a Spectral-Galerkin Approximation to the Helmholtz Equation in Exterior Domains

“…[12,17,18]) in this direction relies on the explicit form of Green's function which is very difficult, if not possible, to extend to more general cases. Our proof is based on an argument in [21,10] (see also [30,9]). More precisely, we take two test functions v = u, (r − a)∂ r u ∈ X in (3.13) successively to obtain a priori estimates without using Green's functions.…”

“…In particular, it has been shown, at least for some simple cases, that errors of pth order numerical methods for the Helmholtz equation behave like O(k p+1 h p ) (see, for instance, [18,4,30]). Hence, high-order methods are particularly preferable for this type of problem over low-order methods.…”

“…The main difficulty here is to obtain error estimates with explicit dependence on the wave number. Among the very few results available in this regard are those in [18,30], where the Helmholtz equation in bounded domains with a first-order approximation to the radiation boundary condition was considered and error estimates with explicit dependence on the wave numbers were derived. To the authors' best knowledge, there seems to be no rigorous error estimate available with explicit dependence on the wave number for a numerical scheme to bounded obstacle scattering.…”

The Tailored Finite Point Method

“…[35,36,54]). Then we approximate the coe cient ( ) by a piecewise constant function, i.e., we introduce a function…”

“…In recent decades, many scientists presented e cient methods for this class of problems with constant coe cients, including the discrete singular convolution method [2], the hybrid numerical asymptotic method [13], the spectral approximation method [54], the element-free Galerkin method [59,63], the so-called ultra weak variational formulation [34], and the hybrid numerical-asymptotic boundary integral method [8]. In general, these methods need the restriction ℎ = O(1) for the mesh size ℎ in the simulation.…”