Mróz presents methods for solving the elliptic second order mixed boundary value problem and gives some examples. The original problem is replaced by subproblems on subregions and an auxiliary problem corresponding to the nodes from partition curves. The subproblems are solved directly, while the auxiliary problem is handled by a conjugate gradient method. The paper discusses the rate of the convergence of the conjugate gradient method for the cases when the Neumann and Dirichlet boundary conditions change at points belonging to intersection curves.