An iterative method is presented for solving plane strain and plane stress problems for homogeneous and isotropic elastic materials. Displacements or some combination of displacements and stresses are prescribed on the boundary of the elastic solid. The iterates are evaluated numerically by difference methods. The direct block factoring method is used to solve the resulting system of algebraic equations. Applications to specific problems are given. A proof of the convergence of the analytic iterations is given for problems where the displacements are specified on the entire boundary.