[petsc-users] The condition number of the ill-conditioned matrix

[Jed Brown]

On Mon, Aug 20, 2012 at 7:07 AM, w_ang_temp <w_ang_temp at 163.com> wrote:

> Hello,
>     I have a problem about the ill-conditioned matrix. I deal with the
> soil-water coupled problem in
> geotechnical engineering. It can be demonstrated that the stiffness matrix
> of soil-water coupled problem is
> ill-conditioned due to the coupled effects of large relative differences
> in stiffnesses and permeabilities
> of materials. But it should not be ill-conditioned for soil-water
> uncoupled problem, although there is no derivation
> in papers.
>     In my three models, the condition numbers are 3.5595E+05, 1.8849E+06,
> 9.0206E+07 for soil-water
> coupled models and 5.2885E+04, 3.3826 E+05, 3.4565E+05 for soil-water
> uncoupled ones, respectively.
> Therefore, in my opinion, the uncoupled models are also ill-conditioned
> due to the large condition
> numbers.
>     However, when using GMRES, CG and CGS, the three models of uncoupled
> ones show rapid convergent rate
> while coupled models have bad convergence performance.
>

This indicates that the spectrum of the simpler model has a small number of
outliers. It could be as simple as boundary condition scaling or it could
be more meaningful. The coupled model does not have that nice property, so
convergence is slower. You should start with:

1. non-dimensionalize: scale your model to use some representative units so
that the solution and residual are well-scaled

2. find an effective preconditioner. PCFIELDSPLIT might work well, but you
have to understand methods to know how to drive it. Look at the examples
and read through the literature in your field to find something you want to
implement, then ask here if you need suggestions for how to implement. (It
can often be done using only command line options.)
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