Hello, I am looking for an efficient, accurate (2nd order) and stable (inf sup) discretization of the incompressible Navier-Stokes equations. A good thing would be a kind of comparison of several discretizations. Thanks a lot. Andreas

It is a difficult task for incompressible Navier-Stokes equations. You can try projection method,staggered grid finite difference, implicit in convective and viscous terms, dropping cross-derivative to simplify. Even so, when solving the resulted algebraic eq, some approximation has to be made, which hardly ensure table (inf sup). However, if a steady-state is desired, artifical compressibility method in implicit scheme renders larger CFL (can be 500 in 3D).

If your problem is in two dimensions and you are starting from scratch, the simplest discretization uses divergence-free Hermite finite elements derived as the curl of a particular cubic element with stream function and solenoidal velocity components as degrees of freedom. This is related to an elasticity element that was developed for plates many years ago. While defined on rectangles, the element can easily be generalized to elements with curved boundaries. Since the divergence condition is built in, there is no continuity equation or inf-sup condition to worry about.

In three dimensions there is a linear Hermite element. There are no quadratic Hermite elements known so far, although there is a Lagrange element with constant divergence which makes the continuity condition easy to implement.