2024 , Volume 29, № 6, p.93-107

The purpose of the paper is to experimentally find the domain of optimal parameters of the constructed numerical method for solving non-stationary Navier–Stokes equations in a polygonal domain with a corner singularity, i.e. in a domain with reentrant corner on the boundary. Time discretization is carried out using Runge–Kutta schemes of the 1st and 2nd order. At each moment of time, the solution of the problem is defined as Rν-generalized one in special sets of weight spaces. Unlike the classical variational formulation, we obtain a nonsymmetrical one. We use the 2nd order Runge–Kutta scheme in the case of its strong L-stability or the 1st order scheme due to the established fact associated with the preservation of the energy balance of the approximation velocity field for a certain Rν-generalized solution of the problem. A numerical method that does not reduce the convergence rate of the approximate solution to the exact one at each time instant without condensing the mesh in the vicinity of the singularity point is proposed. In this case, this order of convergence does not depend on the value of the reentrant corner and is equal to one relative to the grid step. Actually, using the classical finite element method and the finite difference method, the error arising in the vicinity of the reentrant corner extends to the entire computational domain, even where the solution has the necessary smoothness. As a result, contrary to our approach, the order of convergence decreases. Moreover, greater value of the reentrant corner leads to the greater difference between approaches in terms of the approximation order. The proposed method is based on the introduction of a weight function to some extent in the variational formulation of the problem and special basis functions in the finite element method.