Lucas,
I'm trying to precondition my system which can be solved with GMRES
(with increasing iteration number for increasing size) but the standard
preconditioners are either increasing the number of iterations, or
causing the solver not to converge.
Preconditioner design is difficult. This is why I recorded so many
lectures on it :-)
The best approach to solving block systems *efficiently* is to use block
preconditioners. They can have multiple levels of Schur complementing --
in each step, you reduce the size of the problem by one block row and
column. You can also call a 2x2 part of the matrix a block in itself --
for example, for your matrix you might consider splitting it as
[B X]
[Y A]
where
X = [C D]
Y = [L_psi, 0]^T
A = [M_chi, 0; L_chi M_phi]
Then you apply the Schur complementing, which should be relatively
straightforward because A is invertible. In fact, because A is block
triangular, it can easily be solved with by solving for two mass
matrices, each of which is cheap. Then you'd come up with a
preconditioner in the same way as discussed in the "Possibilities for
extensions" of step-22 that uses the fact that you can form a Schur
complement.
Whether that results in a good preconditioner is a separate question,
and one on which it is possible to spend a year or two. But it's
probably worth investigating.
Separately, there is of course also the possibility of using a direct
solver. If you're running in 2d, that would be my first choice -- up to
~200k DoFs, direct solvers are very competitive.
Best
W.
--
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Wolfgang Bangerth email: [email protected]
www: http://www.math.colostate.edu/~bangerth/
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