FICTITIOUS DOMAIN METHOD WITH THE IDEA OF CONJUGATE OPTIMIZATION FOR NON-LINEAR NAVIER-STOKES EQUATIONS

The article considers 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. A computational algorithm has been developed using the fictitious domain method with the idea of conjugate optimization, which allows to build a homogeneous difference scheme in the entire extended domain. To minimize the Lagrange functional, a gradient method is used, which allows us to find the optimal solution by iterative refinement. In this case, it is necessary to calculate the gradient of the Lagrange functional, which leads to the formulation of the conjugate problem. In this article, the method was developed first for the Burgers equation. The model problem shows the effectiveness of using such a modification. Further, the proposed algorithm was developed to solve the Navier-Stokes equation.