On Discrete Projection Methods for the

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 eeciency, 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 nite element discretizations a huge gain in eeciency may be obtained, particularly in the highly nonstationary case. Our theoretical results are connrmed 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 signiicant pressure boundary layers occur, even for the pure projection schemes. These considerations and rst numerical tests in 3D give hope to obtain a powerful CFD{tool. Introduction The eecient solution of the incompressible Navier{Stokes equations, u t ? u + (u r)u + rp = f ; ru = 0 ; in (0; T] ; (1) for given force f and viscosity , with prescribed boundary values on the boundary @ and an initial condition at t = 0, is still one of the big numerical tasks, especially in 3D for long time calculations. From the large variety of existing solution schemes nonlinear, coupled methods of defect correction type (up coupling, see 4],,11],,17]) as well as operator splitting techniques (for instance, SIMPLE or projection methods in combination with quasi{Newton schemes, see 3],,5],,9],,12]) are two classes of most common solution schemes. Both have their pros and cons. The fully coupled solution methods are more attractive from the point of theoretical analysis and stability (large time steps possible because of fully implicit treatment), and the convergence rates are often better in the range of low to middle Reynolds numbers. On the other hand, the numerical eeort required by the corresponding linear algebra tools (block Gauu{ 2 Seidel, Vanka smoother, distributive smoothers in a multigrid approach, …