[petsc-users] replacing Schur complement in pcfieldsplit

[Matthew Knepley]

On Fri, Jun 22, 2012 at 1:46 AM, Klaij, Christiaan <C.Klaij at marin.nl> wrote:

> > > > >
> > > > > In order to implement SIMPLE-type preconditioners for the
> > > > > incompressible Navier-Stokes equations (Elman e.a. JCP 227, 2008)
> > > > > using the PCFieldSplit framework, it looks like I need to replace
> > > > > inv(A00) by some cheap approximation
> > > > >
> > > > > 1) in the Schur complement
> > > > >
> > > >
> > > > When you have a Schur FS, the '0' solver is this approximation.
> > > >
> > > >
> > > > > 2) in the L and/or U of the LDU factorization
> > > > >
> > > >
> > > > You can use whatever PC you want for the solver mentioned above.
> > > >
> > > >
> > > > > 3) while keeping A00 in the D
> > > > >
> > > >
> > > > I think what you want here is -pc_fieldsplit_real_diagonal.
> > >
> > > Let me get this straight. Looking at the full LDU factorization
> > > of the block matrix. Citing from the manual:
> > >
> > >  For the Schur complement preconditioner if
> > >  J = ( A00 A01 )
> > >      ( A10 A11 )
> > >
> > >  the preconditioner using full factorization is
> > >      ( I   -A10 ksp(A00) ) ( inv(A00)     0  ) (     I          0  )
> > >      ( 0         I       ) (   0      ksp(S) ) ( -A10 ksp(A00)  I  )
> > >
> >
> > Yes.
> >
> >
> > > Clearly inv(A00) occurs four times, right? In L and in U and
> > > twice in D. Now if I somehow overrule the '0' solver with my
> > >
> >
> > Yes
> >
> >
> > > approximation and use -pc_fieldsplit_real_diagonal, the effect
> > > would be that inv(A00) is replaced in L, in U and in S but not in
> > > the 00-block of D?
> > >
> >
> > No. What this says is that we should use the action of the
> > actual matrix rather than the preconditioner matrix in the solver.
>
> Odd. Don't you *always* need the action of the actual matrix (and
> of the preconditioner) in a Krylov subspace method?
>

No.


> >
> > I now think I have a better idea what you want, but it would be
> > helpful if you wrote it out completely in linear algebra notation, as
> > above. Right now, we use the same KSP for all 4 places above.
> > Using different KSPs would require a small code change, which I
> > can make if you give me a better idea what you want.
>
> Maybe there is a mistake in the manual, shouldn't it be
>
>  ( I   -A01 ksp(A00) )
>  ( 0         I       )
>

Yes, the comments in fieldsplit.c are correct.


> in the factorization above instead of -A10 ksp(A00)? SIMPLE-type
> preconditioners are usually written as:
>
>  ( I   -A01 dA00^(-1) ) ( A00 0 )^(-1)
>  ( 0         I        ) ( A10 S )
>
> with S = A11 - A10 dA00^(-1) A01, where dA00^(-1) is the inverse
> of the diagonal of A00. Therefore it only requires one solve for
> A00 and one solve for S.
>

Okay, let me think about it.

    Matt


> >
> >
> > > And what's the function name corresponding to
> > > -pc_fieldsplit_real_diagonal?
> > >
> >
> > We have not put one in yet.
>
> Please let me know when you do.
>
> >
> >    Thanks,
> >
> >       Matt
>
>
> dr. ir. Christiaan Klaij
> CFD Researcher
> Research & Development
> E mailto:C.Klaij at marin.nl
> T +31 317 49 33 44
>
> MARIN
> 2, Haagsteeg, P.O. Box 28, 6700 AA Wageningen, The Netherlands
> T +31 317 49 39 11, F +31 317 49 32 45, I www.marin.nl
>
>


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