On the numerical solution of the direct scattering problem for an open sound-hard arc
A boundary integral equation approach is used to solve the (Neumann) boundary value problem for the Helmholtz equation in \(\mathbb{R}^2 \setminus \Gamma\) modelling the scattering phenomena for time-harmonic acoustic waves by a sound-hard open arc \(\Gamma\). (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 \textit{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. \textit{S. G. Michlin, S. Prößdorf} [Singuläre Integraloperatoren, Akademie-Verlag Berlin (1980; Zbl 0442.47027)]. A numerical example (bowl-shaped open arc) shows exponential convergence.
scientific article; zbMATH DE number 910321
scientific article