A numerical method for the solution of 3D-integral equations of electro-static theory based on Gaussian approximating functions

In this work, numerical solution of 3D-integral equations of electro-static theory for an infinite homogeneous medium with a heterogeneous isolated inclusion is considered. Gaussian approximating functions are used for discretization of the integral equations of the problem. These functions essentially simplify construction of the matrix of the system of linear algebraic equations to which the problem is reduced after discretization. The developed method is meshfree, and only the coordinates of the nodes (centers of the Gaussian functions) in the region occupied by the inclusion are initial data required for the realization of the method. For acceptable precision of the numerical solutions of 3D-problems, the number of nodes (approximating functions) should be sufficiently large, and after discretization, the original integral equation is reduced to a system of linear algebraic equations with a non-sparse matrix of high dimension. The multipole expansion method is proposed for the iterative solution of this system. The comparison of the numerical and exact distributions of electro-static fields inside spherical inclusions with radially changing properties is presented and analyzed.