Analytical-Numerical Method for Some Elliptic Boundary Value Problems with Discontinuous Coefficient in Domains with Polyhedral Corners

The paper considers the development of an analytical-numerical method for solving the Dirichlet–Neumann problem for the equation $$\operatorname{div} (\varkappa \nabla u) = 0$$ in a domain $$\Omega$$ containing a cone with base of general form, including a polyhedral corner. The coefficient $$\varkappa$$ of the equation is piecewise constant with discontinuity along a 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 permits one to obtain not only the 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.