

Preprint 43/2010
Well conditioned boundary integral equations for two-dimensional sound-hard scattering problems in domains with corners
Catalin Turc, Jeffrey Ovall, and Akash Anand
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Submission date: 05. Aug. 2010
Pages: 28
published in: Journal of integral equations and applications, 24 (2012) 3, p. 321-358
DOI number (of the published article): 10.1216/JIE-2012-24-3-321
Bibtex
MSC-Numbers: 65N35, 65R20
Keywords and phrases: acoustic scattering, combined-field integral equations, geometric singularities
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Abstract:
We present several well-posed, well-conditioned integral equation formulations for the
solution of two-dimensional acoustic scattering problems with Neumann boundary conditions in
domains with corners. We call these integral equations Direct Regularized Combined Field Integral
Equations (DCFIE-R) formulations because (1) they consist of combinations of direct
boundary integral equations of the second-kind and first-kind integral equations which are
preconditioned on the left by coercive boundary single-layer operators, and (2) their unknowns are
physical quantities, i.e the total field on the boundary of the scatterer. The DCFIE-R equations
are shown to be uniquely solvable in appropriate function spaces under certain assumptions on
the coupling parameter. Using Calderón’s identities and the fact that the unknowns are bounded
in the neighborhood of the corners, the integral operators that enter the DCFIE-R formulations
are recast in a form that involves integral operators that are expressed by convergent integrals
only. The polynomially-graded mesh quadrature introduced by Kress enables the high-order
resolution of the weak singularities of the kernels of the integral operators and the singularities
in the derivatives of the unknowns in the vicinity of the corners. This approach is shown to
lead to an efficient, high-order Nyström method capable of producing solutions of sound-hard
scattering problems in domains with corners which require small numbers of Krylov subspace
iterations throughout the frequency spectrum. We present a variety of numerical results that
support our claims.