Approximation of Scalar Waves in the Space-Frequency Domain

Abstract

A numerical method for approximating a pseudodifferential system describing attenuated, scalar waves is introduced and analyzed. Analytic properties of the solutions of the pseudodifferential systems are determined and used to show convergence of the numerical method. Experiments using the method are reported.

Spectral Approximation of the Helmholtz Equation with High Wave Numbers

“…We note that error estimates for finite element approximations to the Helmholtz equation (2.1) with high wave numbers were derived in [8,18,19] for the 1-D case, and in [9,6] for 2-D cases and in [12] for the 1-D Bessel equation reduced from a 3-D spherical domain, respectively.…”

Least-Squares Spectral Collocation Parallel Methods for Parabolic Problems

“…Then, the Fourier transform p satisfies the following set of elliptic problems depending on ω: for all ω ∈ R The approximate solution for the problem (1.1) was obtained by time stepping methods such as backward Euler and Crank-Nicolson methods traditionally. In recent a natural parallel algorithm which does not require any significant communication costs was introduced by transforming the parabolic problem (1.1) in the space-time domain into the independent elliptic problems (1.2) in the space-frequency domain [11,12,15,20,21]. See [22,23,27,28] for the Laplace transformation.…”

An Elliptic Regularity Coefficient Estimate for a Problem Arising from a Frequency Domain Treatment of Waves

“…For co = 0, (1.2) becomes a Neumann problem and the necessary and sufficient condition for the existence and uniqueness (up to an additive constant) of the solution w(-, 0) is that (1.3) [ f(x,0)dx = 0; Ja (1.3) will be assumed to hold, though the case co -0 will not be of interest in this paper. For co > 0, there exists a unique solution u(-, co) £ HX(Q) for a given /(•, co) £ H~X(Q); see [4]. Denote the solution operator for (1.2) by T(co) : L2(Q) -» H2(Q), so that u(-, co) = T(co)f(-, co).…”