Application of boundary integral equation method to numerical solution of elliptic boundary-value problems in $\mathbb{R}^3$

In this paper we propose numerical methods for solving interior and exterior boundary-value problems for the Helmholtz and Laplace equations in complex three-dimensional domains. The method is based on their reduction to boundary integral equations in $\mathbb{R}^2$. Using the potentials of the simple and double layers, we obtain boundary integral equations of the Fredholm type with respect to unknown density for Dirichlet and Neumann boundary value problems. As a result of applying integral equations along the boundary of the domain, the dimension of problems is reduced by one. In order to approximate solutions of the obtained weakly singular Fredholm integral equations we suggest general numerical method based on spline approximation of solutions and on the use of adaptive cubatures that take into account the singularities of the kernels. When constructing cubature formulas, essentially non-uniform graded meshes are constructed with grading exponent that depends on the smoothness of the input data. The effectiveness of the method is illustrated with some numerical experiments.