On Wed, Aug 14, 2013 at 3:56 PM, Umut Tabak <[email protected]> wrote:
> On 08/14/2013 10:37 PM, Jed Brown wrote: > >> Umut Tabak <[email protected]> writes: >> >> Dear all, >>> >>> I am looking at a system where I am trying to investigate this >>> ill-conditioned problem with some iterative tricks or not. Namely, the >>> system that I try to solve is >>> >>> (B - C^T A^{-1}C) x2 = b2 >>> >>> which results from block symmetric representation >>> >>> A C >>> C^T B >>> >> What physics do you have here? >> > Hi Jed, > > 'A' results from the discretization of structural field equations which is > also ill-conditioned. More specifically, it is (Ks-a*Ms) where Ks and Ms > are stiffness and mass matrices of the structural domain. > However, 'B' results from the discretization of the Helmholtz operator for > the fluid domain. It is also similarly represented as (Kf-a*Mf) as above. > Okay, this is fluid-structure interaction. Why not start with the multiplicative combination in PCFIELDSPLIT first? Then you can move to Schur complement with just an option if you figure out a good preconditioner? Matt > >> Both CG and MINRES require an SPD preconditioner. It sounds like B is a >> poor approximation to the Schur complement S = B - C^T A^{-1} C. >> Depending on your application area, there are a few classes of >> preconditioners that you might consider. These include the >> least-squares commutator, physics-based approximate commutator, >> SIMPLE(R), and DD and multigrid methods applied directly to the >> indefinite problem. >> > Unfortunaltely, yes, even if I have the complete factor for B and even if > C is a pretty sparse matrix, this is not a good preconditioner eventually, > that is clear to me as well. > > Before leaving these ideas, I am trying to convince myselft that this idea > is not useful and cannot be improved further. But, as a poor engineer ;), I > had the feeling that since the fluid part only includes one variable which > is the pressure and the domain is homogeneous, I would expect some better > ways to exist in order to solve this problem. > > Since the domain is homogeneous, at least the fluid domain, and it is > modelled with a scalar variable, I was thinking that scaling should not be > a problem. > > But, there is another important point, due to the modelling approach used > Kf is a singular matrix with one zero eigenvalue(and this is always the > case for a specific type of boundary condition which is the hard wall > condition) and Mf is pretty well conditioned as a standalone matrix. The > source of the problem is writing representing B as (Kf-a*Mf) or as > (Kf/a-Mf) in the original block diagonal representation. > > Can you figure out something more after these explanations? What would you > suggest as a first try and, maybe, a couple of more? > Thanks, > Umut > -- What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead. -- Norbert Wiener
