Analysis of the mixed boundary value problem for the Poisson's equation

The mixed boundary value problem for the Poisson's equation is examined in a bounded flat domain. The problem is continued in a variational form through the boundary with the Dirichlet condition to a rectangular domain. To solve the continued problem, a modified method of fictitious components in a variational form is formulated. The continued problem in a variational form is considered on a finite-dimensional space. To solve the previous problem, a modified method of fictitious components on a finite-dimensional space is formulated. To solve the continued problem in matrix form, the known method of fictitious components is considered. It is shown that in the method of fictitious components the absolute error in the energy norm converges with the speed of a geometric progression. To generalize the method of fictitious components, a new version of the method of iterative extensions is proposed. The continued problem in matrix form is solved using the method of iterative extensions. It is shown that in the proposed version of the method of iterative extensions, the relative error converges in a norm that is stronger than the energy norm of the problem with a geometric progression rate. The iterative parameters in the specified method are selected using the minimum residual method. The conditions which are sufficient for the convergence of the applied iterative process are indicated. An algorithm which implements the proposed version of the method of iterative extensions is written. In this algorithm, an automated selection of iterative parameters is conducted, and the stopping criterion is established when achieving an estimate of the required accuracy. An example of the application of the method of iterative extensions for solving a particular problem is given. In the calculations, the condition for achieving an estimate of the relative error in the norm that is stronger than the energy norm of the problem is set. However, the relative errors of the obtained numerical solution of the example of the original problem are shown in other ways. For example, the relative error in grid nodes is calculated pointwise. To achieve a relative error of no more than a few percent, just a few iterations are required. Computational experiments confirm the asymptotic optimality of the method obtained in theory.