Justification of the collocation method for the integral equation for a mixed boundary value problem for the Helmholtz equation
For a bounded domain in \(\mathbb R^3\) with smooth boundary \(S\), the Helmholtz equation is considered along with a third kind boundary condition at \(S\) and the Sommerfeld condition at infinity. Using the fundamental solution and potentials, this problem is rewritten as the determination of a density on \(S\). Proposing appropriate cubature formulas and investigating their accuracy, a discrete problem arises the existence and unicity properties of which are shown. The convergence of the method is analysed invoking a result of \textit{G. M. Vajnikko} [J. Sov. Math. 15, 675--705 (1981; Zbl 0582.65046); translation from Itogi Nauki Tekh., Ser. Mat. Anal. 16, 5--53 (1979)]. The convergence order is shown to depend on a distance characterizing the discretization of \(S\), on a Hölder exponent connected with \(S\), and on the modulus of continuity of the function on the right hand side of the boundary condition. The result generalizes earlier work of \textit{F. A. Abdullaev} and the author [Differ. Equ. 40, No. 1, 89--93 (2004; Zbl 1070.65123); translation from Differ. Uravn. 40, No. 1, 82--86 (2004)]. The investigation is carried out for any wave number but the influence of the latter on the estimates is not shown.
scientific article; zbMATH DE number 6700381