A numerical approach for solving one nonlinear problem of hydrodynamics in a non-convex polygonal domain

Abstract

The paper considers the two-dimensional stationary problem obtained as a result of time discretization of the nonlinear Navier-Stokes equations in rotation form, which describe the incompressible viscous fluid flow in a non-convex polygonal domain. In order to solve a nonlinear problem, it is necessary to find solutions to a sequence of approximate linear problems. For linear problems, the concept of an Rv-generalized solution in weighted sets obeying the weighted analogue of the Ladyzhenskaya-Babuška-Brezzi condition is introduced. A scheme of the weighted finite element method is constructed which suppresses the error arising in the neighborhood of the vertex of the reentrant angle and does not allow it to propagate into the inner part of the computational domain. The mass conservation law is valid directly at the grid nodes, and not only in the (weak) integral sense. Numerical experiments are carried out and a comparative analysis is made in polygonal domains with different values of reentrant angles. The advantage of the proposed approach over the classical finite element method is the order of convergence with respect to the grid step. A set of optimal parameters of the method is determined to obtain the required order of convergence. In this case, the exponent of the weight function depends on the value of the reentrant angle, and the parameters determining the Rv-generalized solution are independent on its value. In contrast to the classical approach, the order of convergence of the proposed approximate method to the exact solution of a nonlinear problem does not depend on the value of the reentrant angle. The constructed approximate method can be used without geometric mesh refinement in the vicinity of the singularity point.