On discrete projection methods for the incompressible Navier-Stokes equations: an algorithmical approach

Abstract

We derive a general class of iteration schemes for the incompressible Navier-Stokes equations which contains fully coupled solution techniques as well as operator splitting/projection methods. We combine the advantages of both, namely accuracy/ stability and efficiency, and obtain a special form of discrete projection schemes. In combination with a nonlinear iteration of quasi-Newton type one may use these schemes analogously to the well known pure projection schemes, e.g. of Chorin and Van Kan, or apply them as preconditioners in a defect correction approach to obtain the fully coupled Galerkin solution. The corresponding complexity analysis shows that in combination with certain nonconforming finite element discretizations a huge gain in efficiency may be obtained, particularly in the highly nonstationary case. Our theoretical results are confirmed by comparative numerical tests for both types of schemes. It turns out that the appropriate time steps for the pure projection approach are only moderately smaller than those for the fully implicitly coupled schemes, but that the work to obtain comparative results with the discrete projection methods as solvers is much lower. An interesting observation is that in the case of higher Reynolds numbers no significant pressure boundary layers occur, even for the pure projection schemes. These considerations and first numerical tests in 3D give hope to obtain a powerful CFD-tool.