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On the numerical solution of the direct scattering problem for an open sound-hard arc. (English) Zbl 0854.65106
A boundary integral equation approach is used to solve the (Neumann) boundary value problem for the Helmholtz equation in 2 Γ modelling the scattering phenomena for time-harmonic acoustic waves by a sound-hard open arc Γ. (Such problems arise also in the analysis of cracks.) Using the so-called cosine substitution the integral equation is found to be essentially the same as that for a closed boundary, considered e.g. by R. Kress [J. Comput. Appl. Math. 61, No. 3, 345-360 (1995; Zbl 0839.65119)]. Hence, the integral equation is solved approximately by a quadrature formula method, and error estimates in Hölder norms are found by standard techniques, cf. S. G. Michlin, S. Prößdorf [Singuläre Integraloperatoren, Akademie-Verlag Berlin (1980; Zbl 0442.47027)]. A numerical example (bowl-shaped open arc) shows exponential convergence.
MSC:
65N38Boundary element methods (BVP of PDE)
76Q05Hydro- and aero-acoustics
76M15Boundary element methods (fluid mechanics)
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation