On uniform convergence of semi-analytic solution of Dirichlet problem for dissipative Helmholtz equation in vicinity of boundary of two-dimensional domain

Abstract

In the framework of the collocation boundary element method, we propose a semi-analytic approximation of the double-layer potential, which ensures a uniform cubic convergence of the approximate solution to the Dirichlet problem for the Helmholtz equation in a two-dimensional bounded domain or its exterior with a boundary of class $C^5$. In order to calculate integrals on boundary elements, an exact integration over the variable $\rho:=(r^2-d^2)^{1/2}$ is used, where $r$ and $d$ are the distances from the observed point to integration point and to the boundary of the domain, respectively. Under some simplifications we prove that the use of a number of traditional quadrature formulas leads to a violation of the uniform convergence of potential approximations in the vicinity of the boundary of the domain. The theoretical conclusions are confirmed by a numerical solving of the problem in a circular domain.