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

[Dave May]

> -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)
>
>
Huh.
If you called PCFieldSplitSetIS(), then certainly the FS knows you have
four fields and you shouldn't need to set the block size 4 option. I know
this is true as I use it all the time. When you start grouping new splits
(in your case u,v,w) I would have also thought that the block size 3 option
would also be redundant - however I use this option less frequently.



> 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.


The exact same analysis applies directly to any stable mixed u-p
discretization (I agree with all Matt's comments as well). If your
stablilization term is doing what is supposed to, then your discretization
should be stable.
I don't know of any papers which show results using your exact
discretization, but here are some preconditioning papers for Stokes which
employ FV discretizations:

@article{olshanskii1999iterative,
  title={An iterative solver for the Oseen problem and numerical
solution of incompressible Navier--Stokes equations},
  author={Olshanskii, Maxim A},
  journal={Numerical linear algebra with applications},
  volume={6},
  number={5},
  pages={353--378},
  year={1999}
}
@article{olshanskii2004grad,
  title={Grad-div stablilization for Stokes equations},
  author={Olshanskii, Maxim and Reusken, Arnold},
  journal={Mathematics of Computation},
  volume={73},
  number={248},
  pages={1699--1718},
  year={2004}
}

@article{griffith2009accurate,
  title={An accurate and efficient method for the incompressible
Navier--Stokes equations using the projection method as a
preconditioner},
  author={Griffith, Boyce E},
  journal={Journal of Computational Physics},
  volume={228},
  number={20},
  pages={7565--7595},
  year={2009},
  publisher={Elsevier}
}

@article{furuichi2011development,
  title={Development of a Stokes flow solver robust to large viscosity
jumps using a Schur complement approach with mixed precision
arithmetic},
  author={Furuichi, Mikito and May, Dave A and Tackley, Paul J},
  journal={Journal of Computational Physics},
  volume={230},
  number={24},
  pages={8835--8851},
  year={2011},
  publisher={Elsevier}
}
@article{cai2013efficient,
  title={Efficient variable-coefficient finite-volume Stokes solvers},
  author={Cai, Mingchao and Nonaka, AJ and Bell, John B and Griffith,
Boyce E and Donev, Aleksandar},
  journal={arXiv preprint arXiv:1308.4605},
  year={2013}
}

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 would say you are trying to solve a saddle point system, only one which
has been stabilized. I expect your stabilization term should vanish in the
limit of h -> 0. The block preconditioners are directly applicable to what
you are doing, as are all the issues associated with building
preconditioners for schur complement approximations. I have used FS based
preconditioners for stablized Q1-Q1 finite element discretizations for
Stokes problems. Despite the stabilization term in the p-p coupling term,
saddle point preconditioning techniques are appropriate. There are examples
of this in src/ksp/ksp/examples/tutorials - see ex43.c ex42.c





> 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).
>

Oh yes, you are right, that is what is done in this example. But you should
note that this is not the default way petsc's fieldsplit preconditioner
will define the schur complement \hat{S}.  This particular piece of code
lives in the users example.

If you really wanted to do this, the same thing could be configured on the
command line:
  -XXX_ksp_type preonly -XXX_pc_type ksp



> >
> > 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
> >
> >
>
>
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.mcs.anl.gov/pipermail/petsc-users/attachments/20150206/4eeef0ff/attachment.html>