ON UNIFORM CONVERGENCE OF SEMI-ANALYTIC SOLUTION OF DIRICHLET PROBLEM FOR DISSIPATIVE HELMHOLTZ EQUATION IN VICINITY OF BOUNDARY OF TWO-DIMENSIONAL DOMAIN

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.