How to implement a block version of MatDiagonalScale?

[Stephane Aubert <stephane.aubert@xxxxxxxxxxx>]

Hello,
I'm "playing" with badly conditioned block matrices (MPIBAIJ type) 
arising from turbulent compressible fluid dynamics (RANS eqs, block size 
= 5 (laminar) or 7 (turbulent)).
I tested that using bi-normalization approach to build l and r vectors, 
then calling MatDiagonalScale(A,l,r), improve the accuracy and the 
convergence of GMRES+ILU(0), basically by reducing the dependence to the 
mesh cells size and to the variables magnitudes (in particular for k-w 
turbulence model).
Now, I would like to go further and use L and R block diagonal matrices, 
instead of vectors, for example to get block identity along the diagonal 
of L.A.R. The sparcity of A is preserved, from a block point of view.
My first idea is to use a LU factorization of Aii, with L=Li^(-1), 
R=Ui^(-1), Aii=Li.Ui.

My question is: How to compute L and R using PETSC available functions 
in a clever way, instead of calling some LAPACK functions after getting 
individual diagonal blocks in my own piece of code?

My actual guess is:

  1. Create a new matrix D with MPIBDIAG type, from the block-diagonal
     of A using MatGetSubMatrices(), to do something like MatGetDiagonal().
  2. Create a PC of type PCLU using D as operators, with the sequence
     PCCreate/PCSetType/PCSetOperators/PCSetUp.
  3. Get Li and Ui. This is where I'm glued: I can't figure out how to
     use PCGetFactoredMatrix() to get Li and Ui... Is MatLUFactor() a
     better candidate?
  4. Do I need to compute Li^(-1) and Ui^(-1), or is there some
     "backsubstitution" functions available?

With many thanks for your suggestions,
Stef.

--
___________________________________________________________
Dr. Stephane AUBERT, CEO & CTO
FLUOREM s.a.s
Centre Scientifique Auguste MOIROUX
64 chemin des MOUILLES
F-69130 ECULLY, FRANCE
International:      fax: +33 4.78.33.99.39      tel: +33 4.78.33.99.35
France:             fax: 04.78.33.99.39         tel: 04.78.33.99.35
email: stephane.aubert@xxxxxxxxxxx
web: www.fluorem.com