An efficient quasi-linear finite element method for solving the incompressible Navier-Stokes equations at large Reynolds numbers

A numerical method is described for solving the time-dependent Navier-Stokes equations for incompressible viscous fluid flows. The method results from a three step implicit scheme for the time discretization and from a finite element approximation for the space discretization. The typical fractional step method leads to an uncoupling between the nonlinearity and incompressibility and the convective terms appear in a linearized semi-implicit form. The discretization avoids the use of Least Squares methods to treat the nonlinearity. Excellent convergence properties are obtained which improve as the Reynolds number increases. Conjugate gradients algorithms are developed for solving the resulting linear systems, including unsymmetrical matrices. For convection dominated flows, a balancing dissipation technique is introduced in order to stabilize the oscillatory nature of the solution.