Method of fundamental solutions for a conductivity problem

A numerical solution of the interior Dirichlet problem for the homogeneous conductivity equation is considered. After introducing certain assumptions and discretization of the domain, the boundary value problem for a second-order elliptic equation with variable coefficients is reduced to a set of coupled problems with the Helmholtz-type equation defined on multiple disjoint subdomains. The coefficients in the transmission coupled problems are numerically approximated using the Gauss-Legendre and the trapezoid quadrature rules. The coupled problems are simultaneously solved using the method of fundamental solutions, in which the unknown functions are approximated by linear combinations of fundamental solutions, and the coefficients are determined using the collocation method. The applicability and efficiency of the proposed approach are confirmed by the results of numerical experiments.