Асимптотически оптимальное решение модельной задачи для эк­ра­ни­ро­ван­но­го уравнения Пуассона

Abstract

The screened equation is considered on rectangular area, with the mixed boundary conditions. At the numerical solution of this task it is suggested to use iterative factorization after fictitious continuation of the discrete task approximating the solved task. As a result, the solution is based on the solution of systems of the linear algebraic equations with matrixes of the triangular type, which contain no more than three nonzero elements in every line. At rather small error of approximation of the considered task, the demanded relative error of the suggested iterative process is reached in over the number of iterations independent of the discretization parameters. The iterative process turns out to be the method giving an optimal asymptotics as per the number of operations in arithmetic actions. The developed iterative process is based on the characteristic specifics of the stated model task. This task can be obtained in the methods of fictitious components, spaces, when boundary problems for the elliptic equations in areas of a complex shape are being solved. The algorithm is given for the fulfillment of the iterative method with choosing of the iterative parameters in the automatic mode, by applying the minimal residual, corrections method. This gives a criterion to stop the iterative process when the specified preliminarily relative error is obtained. An elementary test example is given on the computing experiments confirming the asymptotic optimality for the iterative method in the number of computing expenses. The fulfillment of the method is substantially based on the use of complex analysis.