University of Magdeburg, Germany

For the numerical solution of the Navier-Stokes equations, first order upwind-methods have shown to be very stable discretizations which produce reasonable solutions up to high Reynolds numbers. Moreover, the corresponding algebraic systems have good properties (M-matrix) that allow to construct robust and efficient multigrid solvers. However, the disadvantage is that the discretization error behaves only like O(h) with respect to the mesh size h such that a very fine mesh is required to get high accuracy of the numerical solution. Therefore, one wants to use higher order discretizations. However, for such discretizations the usual multigrid methods do not perform very well. In this talk, some ways to overcome these problems are presented. By means of numerical experiments, the different approaches are evaluated.