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Abstract
We propose a new robust method for the computation of scattering of highfrequency 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 L2coercive, 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 > infinity for fixed k. We also prove that, as k > infinity, d has to increase only very modestly to maintain a fixed error bound (d similar to k(1/9) is a typical behaviour). Numerical experiments show that the method suffers minimal loss of accuracy as k > infinity, for a fixed number of degrees of freedom. Numerical solutions with a relative error of about 10(5) are obtained on domains of size O(1) for k up to 800 using about 60 degrees of freedom.
Original language  English 

Pages (fromto)  471510 
Number of pages  40 
Journal  Numerische Mathematik 
Volume  106 
Issue number  3 
DOIs  
Publication status  Published  2007 
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Projects
 1 Finished

Boundary Integral Equation Methods for HF Scattering Problems
Graham, I. & Smyshlyaev, V. P.
Engineering and Physical Sciences Research Council
24/03/09 → 23/09/12
Project: Research council