Analytical-numerical method for some elliptic boundary value problems with discontinuous coefficient in domains with polyhedral corners

The paper considers development of an analytic-numerical method for solving the Dirichlet–Neumann problem for the equation $\operatorname{div} (\varkappa \nabla u) =0$ in domain $\Omega$ containing cone with base of general form, including polyhedral corner. The coefficient $\varkappa$ of the equation is piecewise constant with discontinuity along the conical surface, lying in $\Omega$ and having the vertex in common with the original cone. The solution of the problem is represented as the limit of a sequence of linear combinations of functions $\Psi_k$ that make up an approximation system and are constructed explicitly. The method allows to obtain not only a solution in the domain $\Omega$, but also its expansion near the vertex of the cone, and to calculate the corresponding singularity exponents and intensity factors. Some numerical results are presented.