Wang Yuan:
Since I am dealing with a fluid-structure interaction system using a
fully coupled approach.
Degrees of freedom are displacement and pressure.
The Jacobian matrix is block-wise and has the following form：
[ K Q ]
[ Q(T) H ]
where K is the stiffness matrix and has a large condition number.
This is a symmetric matrix, but not guaranteed to be positive definite。
Because the matrix is relatively large, it is hoped that an iterative
solver can be used.
I tried conjugate gradient, GMRES and AMG methods.All of these are
difficult to converge.
Yes, that is not surprising. You need a good preconditioner for these
sorts of problems. One approach is to use
[K^{-1} 0 ]
[0 H^{-1}]
as a preconditioner, and then you can use CG or Minres for the solver
(depending on whether or not the overall matrix is positive definite).
But you can also include a term either in the upper or lower zero block
above, which often makes the preconditioner better; in that case, the
preconditioner is non-symmetric, and you need GMRES as a solver.
We call these kinds of preconditioners "block preconditioners" because
they exploit the block structure of the matrix. step-20 and step-22
first discuss these sorts of issues, and I would encourage you to also
take a look at video lecture 38 at
https://www.math.colostate.edu/~bangerth/videos.html
Best
W.
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Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
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