Fictitious-domain method for the spectral problem for the Stokes operator (Q2714078)

The authors study the spectral problem for the Stokes operator on a domain in \(\mathbb R^3\) with smooth boundary. To solve the problem, the authors use the fictitious-domain method applied to an auxiliary problem in a suitable domain. The coefficients of the operator under consideration are extended by a small parameter \(\varepsilon\) in the so-called fictitious domain, and simultaneously some boundary value conditions are formed on the boundary of the domain.NEWLINENEWLINENEWLINEThe authors prove an existence theorem for the class of weak solutions to the auxiliary problem and establish a priori estimates for the solution obtained. As a result, this approach makes it possible to prove a convergence theorem and find the rate of convergence for eigenfunctions and eigenvalues to a solution of the original problem. This enables us to construct an iteration method for solving Navier--Stokes equations.