On 09/08/2013 06:17 PM, Jed Brown wrote:
block is indefinite, even worse.
A11 indefinite could lead to the Schur complement being singular. The
approach breaks down if that's the case. A11 singular or negative
semi-definite is the standard case.
Even not singular, an already ill-conditioned problem becomes worse with
a Schur complement approach.
And it is almost impossible to define a SPD preconditioner that could
work with S to result in decent iteration counts.
This was a question I asked a couple of weeks ago but did not have the
time to go into details.
I guess these kinds of separation ideas and use of independent
factorizations or combined preconditioners for different blocks will not
work for this problem...
You need to read the literature for problems of your type. There are a
few possible approaches to approximating the Schur complement. The
splitting into blocks also might not be very good.
well I took a look at the literature but could not really find something
useful up until now. The main difference was one of the blocks was
either SPD or well-conditioned(for A00 or A11).
More specifically, my problem is a kind of shifted problem for
eigenvalue solutions without going into too much detail. The symmetric
operator matrix is written as
AA
=
(A-\sigmaB) C
C^T (D-\sigmaE)
and C is a rather sparse coupling matrix
A00 = A-\sigmaB block is ill conditioned due to the shift
and
A11 = (D-\sigmaE)
block is indefinite where D is a singular matrix with one zero
eigenvalue with 1 vector in the null space. E is rather well conditioned
but in combination it is not attackable by iterative methods. Moreover,
Schur complements are defined on this problematic system, AA.
There is some literature on these shifted problems but I am not sure if
I should dive into that field or not at this point.
Trying the 'fieldsplit' approach was some kind of a 'what-if' for me...
But thanks for the help and comments anyway.
