A Dirichlet problem is considered on a rectangle for singularly perturbed linear and quasilinear elliptic equations. When the perturbation parameter equals zero, elliptic equations degenerate into zero-order ones. Special iterative and iteration-free finite difference schemes (in particularly, the schemes using parallel computations) are constructed which converge uniformly with respect to the parameter. Schwarz' method is used to construct the schemes. Necessary and sufficient conditions are given for the solutions of the iterative difference schemes to converge uniformly with respect to the perturbing parameter as the number of iterations increases. It is shown that the use of schemes with parallel computations on multiprocessor computers provides the acceleration of computations.