The Semi-Lagrangian Approximation in the Finite Element Method for the Navier-Stokes Equations

The two-dimensional time-dependent Navier-Stokes equations are considered for a viscous incompressible fluid in a channel. On the outlet boundary, the modified "do nothing" condition is imposed. To construct a discrete analogue, a semi-Lagrangian approximation of the transport derivatives is used in combination with the conforming finite element method for the approximation of other terms. The velocity components are approximated by biquadratic elements and the pressure is approximated by bilinear elements on rectangles. As a result of this combined approximation, the stationary problem with a self-adjoint operator is obtained at each time level. The theoretical results are confirmed by numerical experiments.