[petsc-users] Field Split PC for Fully-Coupled 3d stationary incompressible Navier-Stokes Solution Algorithm

[Matthew Knepley]

On Thu, Feb 5, 2015 at 4:15 PM, Fabian Gabel <gabel.fabian at gmail.com> wrote:

> Thank you for your feedback.
>
> >         -coupledsolve_pc_type fieldsplit
> >         -coupledsolve_pc_fieldsplit_0_fields 0,1,2
> >         -coupledsolve_pc_fieldsplit_1_fields 3
> >         -coupledsolve_pc_fieldsplit_type schur
> >         -coupledsolve_pc_fieldsplit_block_size 4
> >         -coupledsolve_fieldsplit_0_ksp_converged_reason
> >         -coupledsolve_fieldsplit_1_ksp_converged_reason
> >         -coupledsolve_fieldsplit_0_ksp_type gmres
> >         -coupledsolve_fieldsplit_0_pc_type fieldsplit
> >         -coupledsolve_fieldsplit_0_pc_fieldsplit_block_size 3
> >         -coupledsolve_fieldsplit_0_fieldsplit_0_pc_type ml
> >         -coupledsolve_fieldsplit_0_fieldsplit_1_pc_type ml
> >         -coupledsolve_fieldsplit_0_fieldsplit_2_pc_type ml
> >
> >         Is it normal, that I have to explicitly specify the block size
> >         for each
> >         fieldsplit?
> >
> >
> > No. You should be able to just specify
> >
> >
> > -coupledsolve_fieldsplit_ksp_converged
> > -coupledsolve_fieldsplit_0_fieldsplit_pc_type ml
> >
> >
> > and same options will be applied to all splits (0,1,2).
> >
> > Does this functionality not work?
> >
> >
> It does work indeed, but what I actually was referring to, was the use
> of
>
> -coupledsolve_pc_fieldsplit_block_size 4
> -coupledsolve_fieldsplit_0_pc_fieldsplit_block_size 3
>
> Without them, I get the error message
>
> [0]PETSC ERROR: PCFieldSplitSetDefaults() line 468
> in /work/build/petsc/src/ksp/pc/impls/fieldsplit/fieldsplit.c Unhandled
> case, must have at least two fields, not 1
>
> I thought PETSc would already know, what I want to do, since I
> initialized the fieldsplit with
>
> CALL PCFieldSplitSetIS(PRECON,PETSC_NULL_CHARACTER,ISU,IERR)
>
> etc.
> >
> >
>
> >         Are there any guidelines to follow that I could use to avoid
> >         taking wild
> >         guesses?
> >
> >
> > Sure. There are lots of papers published on how to construct robust
> > block preconditioners for saddle point problems arising from Navier
> > Stokes.
> > I would start by looking at this book:
> >
> >
> >   Finite Elements and Fast Iterative Solvers
> >
> >   Howard Elman, David Silvester and Andy Wathen
> >
> >   Oxford University Press
> >
> >   See chapters 6 and 8.
> >
> As a matter of fact I spent the last days digging through papers on the
> regard of preconditioners or approximate Schur complements and the names
> Elman and Silvester have come up quite often.
>
> The problem I experience is, that, except for one publication, all the
> other ones I checked deal with finite element formulations. Only
>
> Klaij, C. and Vuik, C. SIMPLE-type preconditioners for cell-centered,
> colocated finite volume discretization of incompressible
> Reynolds-averaged Navier–Stokes equations
>
> presented an approach for finite volume methods. Furthermore, a lot of
> literature is found on saddle point problems, since the linear system
> from stable finite element formulations comes with a 0 block as pressure
> matrix. I'm not sure how I can benefit from the work that has already
> been done for finite element methods, since I neither use finite
> elements nor I am trying to solve a saddle point problem (?).
>

I believe the operator estimates for FV are very similar to first order
FEM, and
I believe that you do have a saddle-point system in that there are both
positive
and negative eigenvalues.

  Thanks,

     Matt


> >
> >         > Petsc has some support to generate approximate pressure
> >         schur
> >         > complements for you, but these will not be as good as the
> >         ones
> >         > specifically constructed for you particular discretization.
> >
> >         I came across a tutorial (/snes/examples/tutorials/ex70.c),
> >         which shows
> >         2 different approaches:
> >
> >         1- provide a Preconditioner \hat{S}p for the approximation of
> >         the true
> >         Schur complement
> >
> >         2- use another Matrix (in this case its the Matrix used for
> >         constructing
> >         the preconditioner in the former approach) as a new
> >         approximation of the
> >         Schur complement.
> >
> >         Speaking in terms of the PETSc-manual p.87, looking at the
> >         factorization
> >         of the Schur field split preconditioner, approach 1 sets
> >         \hat{S}p while
> >         approach 2 furthermore sets \hat{S}. Is this correct?
> >
> >
> >
> > No this is not correct.
> > \hat{S} is always constructed by PETSc as
> >   \hat{S} = A11 - A10 KSP(A00) A01
>
> But then what happens in this line from the
> tutorial /snes/examples/tutorials/ex70.c
>
> ierr = KSPSetOperators(subksp[1], s->myS, s->myS);CHKERRQ(ierr);
>
> It think the approximate Schur complement a (Matrix of type Schur) gets
> replaced by an explicitely formed Matrix (myS, of type MPIAIJ).
> >
> > You have two choices in how to define the preconditioned, \hat{S_p}:
> >
> > [1] Assemble you own matrix (as is done in ex70)
> >
> > [2] Let PETSc build one. PETSc does this according to
> >
> >   \hat{S_p} = A11 - A10 inv(diag(A00)) A01
> >
> Regards,
> Fabian
> >
> >
>
>
>


-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener
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