Fast solution of a model problem for the biharmonic equation

The biharmonic equation in a domain of rectangular shape when boundary conditions are mixed is being considered. Numerical solution of this boundary value problem uses iterative factorization on fictitious continuation after finite-difference approximation of the problem to be solved. Eventually, everything is reduced to solving the linear systems of algebraic equations, the matrices of which are triangular with three or less nonzero elements in lines. If approximation error of the initial problem is sufficiently small, the demanded relative error of the used iterative process gets obtained in several iterations. In this case, the developed iterative method turns out to be the method that has optimal asymptotics by the number of actions in arithmetic operations. The proposed iterative method essentially uses specificities of the obtained model problem. Such a problem can arise in methods of the type of fictious components, regions and spaces, when boundary value problems with elliptic equations in the regions of sufficiently arbitrary shape are being solved. The algorithm at implementation of the iterative process, when the choice of iterative parameters is made automatically using the method of minimal corrections, is given. The criterion for process termination after achieving the preliminarily determined ratio error is specified. Graphic result of a computational experiment that proves the asymptotic optimality of the iterative method in computational outlay is given. Complex analysis gets essentially used when developing the method.