[petsc-users] Ainsworth formula to solve saddle point problems / preconditioner for shell matrices

[Olivier Jamond]

>     Given the structure of C it seems you should just explicitly construct Sp and use GAMG (or other preconditioners, even a direct solver) directly on Sp. Trying to avoid explicitly forming Sp will give you a much slower performing solving for what benefit? If C was just some generic monster than forming Sp might be unrealistic but in your case CCt is is block diagonal with tiny blocks which means (C*Ct)^(-1) is block diagonal with tiny blocks (the blocks are the inverses of the blocks of (C*Ct)).
>
>      Sp = Ct*C  + Qt * S * Q = Ct*C  +  [I - Ct * (C*Ct)^(-1)*C] S [I - Ct * (C*Ct)^(-1)*C]
>
> [Ct * (C*Ct)^(-1)*C] will again be block diagonal with slightly larger blocks.
>
> You can do D = (C*Ct) with MatMatMult() then write custom code that zips through the diagonal blocks of D inverting all of them to get iD then use MatPtAP applied to C and iD to get Ct * (C*Ct)^(-1)*C then MatShift() to include the I then MatPtAP or MatRAR to get [I - Ct * (C*Ct)^(-1)*C] S [I - Ct * (C*Ct)^(-1)*C]  then finally MatAXPY() to get Sp. The complexity of each of the Mat operations is very low because of the absurdly simple structure of C and its descendants.   You might even be able to just use MUMPS to give you the explicit inv(C*Ct) without writing custom code to get iD.

At this time, I didn't manage to compute iD=inv(C*Ct) without using 
dense matrices, what may be a shame because all matrices are sparse . Is 
it possible?

And I get no idea of how to write code to manually zip through the 
diagonal blocks of D to invert them...

Thanks for helping!