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CHAPTER 4. AMG METHODS FOR THE MIXED PROBLEM 50
Figure 4.1 Using the FE-AMG-isomorphism we can associate coarse basis functions with
basis vectors of R m l . Here we have three basis functions for a certain Ql .
Both are especially delicate in the multigrid setting, because if the modes illustrated in
Figures 2.1 and 2.4 occur on a coarse level, then the smoother on the finer level might not
damp them (they have lower frequency than the modes the smoother is intended to reduce)
and the whole iteration might fail. As both terms have a non-standard h-dependence we
try to reproduce this on coarser levels to avoid a ‘flattening’ of the stabilization. Numerical
tests show that for the SUPG term a S in (2.27) it is sufficient to do a simple scaling, i.e.
A Sl+1 = d √
nl
n l+1
Ĩ l+1
l
A Si Ĩ l l+1. (4.1)
The scaling of the element stabilization will be dealt with later (Section 4.1.2).
Another major part of our strategy is to somehow project the relation of the velocity
and pressure unknowns, which is indicated by the specific finite element, to the coarser
levels. This makes it obvious that we will not construct a “black box” method, i.e. a
method where the user just has to feed in the matrix, and the solution is found in optimal
computation time. We try to exploit more information and hope that this will pay off.
We will now construct coarse level systems, which comply with this strategy, for the
conforming linear elements of Section 2.2.1, namely the modified Taylor-Hood element
P 1 isoP 2 -P 1 , the P 1 -P 1 -stab element, and the Crouzeix-Raviart element P nc
1 -P 0 .