Analysis of the problem for the biharmonic equation

For a biharmonic equation, we consider a mixed problem with the main boundary conditions. We continue the original problem along the boundary with the Dirichlet conditions into a rectangular domain. The continued problem is given as an operator equation. The method of iterative extensions is written out in the operator form when solving the continued problem. The operator continued problem is given on a finite-dimensional subspace. The method of iterative extensions is given for solving the operator continued problem on a finite-dimensional subspace. After discretization, the continued problem is written in the matrix form. The continued problem in the matrix form is solved by the method of iterative extensions in the matrix form. It is established that in the cases under consideration the method of iterative extensions has relative errors converging as a geometric progression in a norm stronger than the energy norm of the extended problem. In the applied iterative processes, the iterative parameters are selected on the basis of minimizing the residuals. We give conditions that guarantee the convergence of the iterative processes used. Also, we present an algorithm that implements the method of iterative extensions in the matrix form. The algorithm performs an independent selection of iterative parameters and provides a criterion for stopping if an estimate of the required accuracy is achieved. A computational example of using the method of iterative extensions on a computer is given.