[petsc-users] function name and meaning of -pc_fieldsplit_schur_factorization_type

[Klaij, Christiaan]

> > The manual page 87 states that "there are several variants
> > of the Schur complement preconditioner obtained by dropping some
> > of the terms, these can be obtained with
> > -pc_fieldsplit_schur_factorization_type <diag,lower,upper,full>"
> >
> > 1) What is (will be) the name of the corresponding function?
> >
>
> It looks like PCFieldSplitSetSchurFactorizationType() is missing.

Would be nice to have.

>
>
> >
> > 3) I'm confused about the minus sign in the <diag> form and its
> > motivation. It doesn't come from the factorization. Is the rhs
> > multiplied with a minus sign as well? Why not follow the sign
> > convention of Elman's taxonomy (eq 3 and 4, JCP 227, 2008)?
> >
>
> Elman's convention is rather specific to finite element discretization of
> incompressible flow. He arbitrarily slaps a negative on one of the matrices
> in the system because it represents a certain kind of stabilization. That
> minus sign would be pretty silly if the matrix was actually positive (mixed
> elasticity, finite-time step stiff waves, etc).
>
> The negative sign in the DIAG form appears because the minimal polynomial
> with that preconditioner has degree 3 (for a nonsingular problem) if you
> have a minus sign there. Read Murphy's note:
> http://eprints.maths.ox.ac.uk/1291/1/NA-99-07.pdf
>
> If you skip that minus sign, that property goes away. Also, if you are
> solving Stokes flow or constrained problems in elasticity, that sign
> convention causes the DIAG form to be positive definite which means that
> you could use it with MINRES which was is the main motivation for using
> DIAG in the first place.

Jed, thanks for pointing me to Murphy's note, I didn't realize we
were allowed to choose the sign in this case. It's rather
counter-intuitive to have a preconditioner which is not an
approximate inverse.


dr. ir. Christiaan Klaij
CFD Researcher
Research & Development
E mailto:C.Klaij at marin.nl
T +31 317 49 33 44

MARIN
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