Thanks a lot, this is exactly what I needed！！
On Wednesday, February 8, 2023 at 5:04:02 AM UTC+8 Wolfgang Bangerth wrote:
>
> 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.
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: bang...@colostate.edu
> www: http://www.math.colostate.edu/~bangerth/
>
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