Fast solution of the model problem for Poisson's equation

Poisson's equation in rectangular area under the mixed regional conditions is considered. Its numerical solution by means of iterative factorizations and fictitious continuation amounts to the solution of the systems of linear algebraic equations with triangular matrixes, in which the quantity of nonzero elements in every line is less than three. At rather insignificant error of approximation of the problem under consideration the set relative error of a numerical method is reached by some iterations. The given iterative method is almost a direct method, asymptotically optimum by the number of arithmetic operations. The iterative method is developed for the specified model problem. This problem turns out to be in the methods of fictitious components at the solution of boundary problems for elliptic differential equations of the second and fourth orders in flat areas. The algorithm for realization of a numerical method with an automatic choice of iterative parameters on the basis of a method of the fastest descent is offered. The criterion to stop an iterative process is set at the achievement of the set relative error of the solution. The graphic results of computing experiments confirming an asymptotic optimality of the method on computing expenses are given. The developing of the method is based on the use of the complex analysis.