Preconditioners for the incompressible Navier Stokes equations

Computational Fluid Dynamics is frequently used nowadays to understand the flow in rivers, blood veins, around cars and planes, etc. This tool can also be used to make better cars and planes and to design dams and dikes to protect against flooding. In this talk we consider simulation with the incompressible Navier Stokes equations. After discretization by the Finite Element Method and linearization (using Picard or Newton-Raphson) one obtains a large sparse linear system. Due to the incompressibility constraint a large zero block appears on the main diagonal of the system matrix. This type of problems is also known as a saddle point problem. Straightforward application of direct or iterative solvers leads to breakdown or divergence of the methods. For large 3 dimensional problems only iterative methods are feasible due to large time and memory requirements of direct solvers.