Variational methods for constructing iterative algorithms.

In this paper, we consider the study of the extreme problem of the fictitious domain method based on the use of the Lagrange functional, with a multiplier defined on the actual boundary and associated with genuine boundary conditions. The theorem of the existence of a generalized solution of the auxiliary FDM problem is proved. The convergence of the solution of the auxiliary FDM problem is shown. For mathematical research, the theory of functional spaces and the technique of obtaining a priori estimates in functional spaces are used. To verify the effectiveness of the developed method, numerical analysis and comparison with other methods of solving the problem were carried out. [ABSTRACT FROM AUTHOR]