FieldSplit Schur preconditioner

[Jed Brown]

I'm happy to see this preconditioner going in.

  (I   -B inv(A)) ( inv(A)   0    ) (I         0  )
  (0    I       ) (   0    inv(S) ) (-C inv(A) I  )

  S = D - C inv(A) B

There are some modifications that greatly improves the performance in my
experience.

First, the `inv(A)' appearing in S need not be the same as the `inv(A)'
appearing explicitly in the preconditioner.  I notice that these are
required to be the same in the current implementation.  For instance, 1
V-cycle of AMG works great inside of S, but I find it is frequently
better to make the outer `inv(A)' somewhat stronger (this is relatively
cheap if S is not very well conditioned).

Also, I have experimented with mixing the full order (matrix-free using
MatShell) operator with low-order preconditioning blocks.  That is, if
the global problem is

J = (A  B)
    (C  D)

P = (Ap  Bp)
    (Cp  Dp)

defining the Schur complement as

  S = D - C inv(A) B

instead of the current

  S' = Dp - C inv(Ap) B

where inv(A) is solved using Ap for preconditioning [1].  Note that in
practice, I usually use -schur_A_ksp_type preonly, but this allows the
more restrictive formulation to fall out as a special case.

It is probably important to make it possible to define S in terms of
either D or Dp.  In my case, I have D available with a fast matrix-free
application, but with MatMFFD it is much more expensive, hence Dp may be
preferable.

[1] Bp,Cp,Dp are not used in this case, though perhaps you would use Dp
to precondition S (in this case Dp is not an approximation of D which is
often singular).  It's kind of an awkward hack, but an alternative is for
assembly code to

  PetscObjectCompose((PetscObject)Dp,"schur_pc",(PetscObject)Sp)

where Sp is the mass matrix or whatever.  This way other
preconditioners would ignore it, but PC_COMPOSITE_SCHUR would use this
matrix if it was available.

Just my thoughts, thanks for working on this.

Jed
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