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On the numerical solution of a hypersingular integral equation in scattering theory. (English) Zbl 0839.65119
The paper studies the hypersingular integral equation arising from the combined double- and single-layer ansatz for the exterior Neumann problem of the two-dimensional Helmholtz equation in smooth domains and a fully discrete method to its numerical solution. The corresponding integral operator is a compact perturbation of the hypersingular operator of the Laplace equation and invertible between Hölder spaces. Thus the trigonometric collocation converges and by using quadratures for the kernels of the perturbation integrals based on trigonometric interpolation a fully discrete method is obtained. It is proved that the convergence rate of the method is determined by the trigonometric interpolation error of the right-hand side, which leads to exponential convergence rates for analytic boundaries and analytic data.

##### MSC:
 65N38 Boundary element methods (BVP of PDE) 35J05 Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation 45E05 Integral equations with kernels of Cauchy type 65R20 Integral equations (numerical methods) 65N12 Stability and convergence of numerical methods (BVP of PDE)