Studying the correctness of boundary problems for Navier–Stokes equations in primitive variables

The Galerkin finite element method was implemented within the framework of the symbolic computation system. This provides studying the correctness of boundary problems for the incompressible viscous flow both numerically and analytically. An approach based on the coupled solution of the Navier–Stokes equations in primitive variables was used. In the problems with the given velocity on boundaries such technique leads to the singular system of linear equations and to impossibility to obtain the solution. The system matrix have zero as multiple eigenvalue. It has been shown that this effect is caused by the solenoidality condition for the velocity field. A regularization approach with a parameter having the physical meaning is also tested. In this case the spectrum contains only one zero, and nonlinear solutions corresponding to experimental data was easily obtained. The boundary problems with the given pressure drop are correct. The Galerkin finite element method for regularized equations is free from scheme viscosity, and the solutions do not depend on the parameters of grids. In commonly used finite-difference methods the different scheme viscosity virtually serves as an implicit regularization parameter, and that results in incommensurability of calculations results.