[petsc-users] [PATCH 2/2] Call KSPSetFromOptions before setting fieldsplit-specific options

[Umut Tabak]

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.