Numerical solution of Laplace's equation on non-simply connected regions on R-2

We consider the interior Dirichlet problem for Laplace's equation on a non- simply connected two-dimensional regions with smooth boundaries. The soluti on is sought as the real part of a holomorphic function on the region, give n as Cauchy-type integral. The approximate double layer density function is found by solving a system of Fredholm integral equations of second kind. B ecause of the non-uniqueness of the solution of the system we solve it usin g a technique based on the solution of the "Modified Dirichlet problem". Th e Nystrom's method coupled with the trapezoidal rule is used as numerical i ntegration scheme. The linear System derived from the integral equation is solved using the conjugate gradient applied to the normal equation. Theoret ical and computational details of thr method are presented.