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 div ( ∇ u) = 0 in a domain Ω containing a cone with base of general form, including a polyhedral corner. The coefficient of the equation is piecewise constant with discontinuity along a conical surface lying in Ω 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 Ψₖ that make up an approximation system and are constructed explicitly. The method permits one to obtain not only the solution in the domain Ω 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.