Víctor Domínguez

A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering

We propose a new robust method for the computation of scattering of high-frequency acoustic plane waves by smooth convex objects in 2D. We formulate this problem by the direct boundary integral method, using the classical combined potential approach. By exploiting the known asymptotics of the solution, we devise particular expansions, valid in various zones of the boundary, which express the solution of the integral equation as a product of explicit oscillatory functions and more slowly varying unknown amplitudes. The amplitudes are approximated by polynomials (of minimum degree d) in each zone using a Galerkin scheme. We prove that the underlying bilinear form is continuous in L2, with a continuity constant that grows mildly in the wavenumber k. We also show that the bilinear form is uniformly L2-coercive, independent of k, for all k sufficiently large. (The latter result depends on rather delicate Fourier analysis and is restricted in 2D to circular domains, but it also applies to spheres in higher dimensions.) Using these results and the asymptotic expansion of the solution, we prove superalgebraic convergence of our numerical method as d → ∞ for fixed k. We also prove that, as k → ∞, d has to increase only very modestly to maintain a fixed error bound (d ∼ k1/9 is a typical behaviour). Numerical experiments show that the method suffers minimal loss of accuracy as k →∞, for a fixed number of degrees of freedom. Numerical solutions with a relative error of about 10−5 are obtained on domains of size

Some properties of layer potentials and boundary integral operators for the wave equation

\mathcal{O}(1)

Dirac delta methods for Helmholtz transmission problems

for k up to 800 using about 60 degrees of freedom.

Algorithm 884: A Simple Matlab Implementation of the Argyris Element

In this paper we propose and analyze composite Filon--Clenshaw--Curtis quadrature rules for integrals of the form

Filon-Clenshaw-Curtis rules for a class of highly-oscillatory integrals with logarithmic singularities

I_{k}^{[a,b]}(f,g) := \int_a^b f(x) \exp(\mathrm{i}kg(x)) \mathrm{d} x

Convergence analysis of a high-order Nyström integral-equation method for surface scattering problems

, where

Full asymptotics of spline Petrov-Galerkin methods for some periodic pseudodifferential equations

k \geq 0

Regularized Combined Field Integral Equations for Acoustic Transmission Problems

,