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

[Jed Brown]

On Fri, Mar 16, 2012 at 08:40, Klaij, Christiaan <C.Klaij at marin.nl> wrote:

> 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.


> 2) It would be nice to know what exactly is being dropped. <full>
> is obvious and apparently <diag> drops the right most matrix, but
> what do lower and upper drop?
>

Lower uses the lower-triangular factor (dropping the strictly upper
triangular factor).


>
> 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.
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