Abstract
In the present work, an improved modification of the fictitious domain method with extension by large coefficients is proposed for a nonstationary nonlinear mathematical model in the Boussinesq approximation describing the motion of an inhomogeneous incompressible fluid in a three-dimensional bounded domain. In the course of justifying the proposed modification of the fictitious domain method, the existence of a weak generalized solution to an auxiliary problem with a small parameter on a global time interval is proved, and it is shown that this solution converges to the weak generalized solution of the original physical problem. Uniform a priori estimates in appropriate functional spaces are obtained for the solutions. For the weak generalized solution, a convergent sequence of approximate solutions is constructed using the Galerkin method. In addition, conditions ensuring the existence of a strong generalized solution on a local time interval and its convergence are formulated. It is shown that, for the proposed modification of the fictitious domain method, the convergence rate of strong generalized solutions is significantly higher than that of the standard fictitious domain method. Within this modification, by an appropriate choice of the parameter, it is possible to obtain a convergence rate required for practical computations.