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.