Krylov Subspace and Multigrid Methods applied to the Incompressible Navier-Stokes Equations C. Vuik, P. Wesseling, and S. Zeng Faculty of Technical Mathematics and Informatics Delft University of Technology Mekelweg 4, 2628 CD Delft, The Netherlands Abstract Krylov subspace and Multigrid methods are two types of promising iterative methods for the solution of large unsymmetric non-diagonally dominant linear systems of algebraic equations resulting from discretization of partial differential equations. We apply these methods to the solution of the incompressible Navier-Stokes equations discretized on staggered grids in general coordinates. The two types of method can both be regarded as powerful techniques to accelerate basic iterative methods, but they have different advantages and disadvantages. For solving non-symmetric problems, a Krylov subspace method called GMRESR has been developed, which is based on GMRES and GCR. An obvious advantage of GMRESR is its easy vectorization, since most of its arithmetic operations are vector updates, vector-vector and matrix-vector multiplications. Furthermore, vector lengths become large as the grid is refined, which improves speed on vector computers. However, the number of iterations grows significantly as the grid gets finer. Multigrid methods do not have this problem. A distinguishing feature of multigrid is that in principle its computational cost is \$O(N)\$, where \$N\/\$ is the number of (discrete) unknowns. But the performance of a multigrid method strongly depends on the smoother used. A simple smoother like point Jacobi is easily vectorizable but not robust. For difficult problems, as in general coordinates, multigrid methods using simple smoothers often fail, and therefore more complicated but robust smoothers are required. A complicated smoother like ILU is robust but hard to vectorize. Furthermore, another problem for multigrid methods is that occurrence of vectors of short length is inevitable, since use of coarser grids is necessary. This hampers multigrid efficiency on vector computers. In this paper, we compare the two types of method on an HP 735 workstation and on a Convex 3840 mini supercomputer. This makes it possible to compare rates of convergence and vectorizability. The foregoing observations on the advantages and disadvantages of the two types of method suggest that combinations of them may be profitable. We specify a combination of a Krylov subspace method with a multigrid method. This combination is applied to the incompressible Navier-Stokes equations and its performances are compared with the original methods.