(Suleyman Demirel University, 2012) Kuttykozhaeva Sh. N.
This paper investigates the method of fictitious domains for solving boundary value problems in hydrodynamics, extending its application to nonlinear elliptic equations. A rigorous justification of the method is presented, and a new approach to achieve the best possible convergence rate from the auxiliary problem to the original problem is proposed. The study demonstrates that as the small parameter tends to zero, the solution of the fictitious domain problem converges to the generalized solution of the original boundary value problem. The theoretical results include existence, uniqueness, and stability estimates, providing a solid mathematical foundation for applying the fictitious domain method in hydrodynamic modeling.