[petsc-users] Rigid body nullspace for Stokes operator

[Amneet Bhalla]

@Mark: Is there some document/paper that I can follow to check the algebra
of these zero eigenvectors/null space modes?

@Jed : We use a projection method preconditioner to solve the coupled
velocity pressure system as described here (
https://urldefense.us/v3/__https://www.sciencedirect.com/science/article/pii/S0021999123004205__;!!G_uCfscf7eWS!ZRrdNNCSkRL0WcX9v_6J47NL0alKPWI62lpOHMD1McnE_Mj0VpDWMYMqnlAYtYcpBM9gXe_PJCQpsqHNGDw-c3gVcg$ ). It is
an approximation of the Schur complement. As a part of projection
preconditioner, we need to solve just the momentum equation separately
without considering the pressure part.

On Tue, Oct 29, 2024 at 8:03 PM Jed Brown <jed at jedbrown.org> wrote:

> And to be clear, we recommend using fieldsplit Schur to separate the
> pressure and velocity part (there are many examples). Applying AMG directly
> to the saddle point problem will not be a good solver because the
> heuristics assume positivity and do not preserve inf-sup stability (nor do
> standard smoothers).
>
> Mark Adams <mfadams at lbl.gov> writes:
>
> > This is linear elasticity and there are 6 "null" vectors (if you removed
> > Dirichlet boundary conditions these are eigenvectors with zero
> eigenvalue):
> > 3 translations, x, y, z, and three rotatiions xx, yy ,zz.
> > x = (1,0,0,1,0,0,1,0 ...)
> > and xx is something like (0, z_1, -y_1, 0, z_2, -y_2, ...) where z_1 is
> the
> > z coordinate of the first vertex, etc.
> >
> > Mark
> >
> > On Tue, Oct 29, 2024 at 3:47 PM Amneet Bhalla <mail2amneet at gmail.com>
> wrote:
> >
> >> Hi Mark,
> >>
> >> Thanks! I am not sure how to construct null space and zero energy modes
> >> manually for this operator. Is there some theory or documentation I can
> >> follow to figure out what the null space and zero energy modes look like
> >> for this operator? Once I know what these are in symbolic form, I think
> I
> >> should be able to construct them manually.
> >>
> >> Best,
> >> --Amneet
> >>
> >> On Tue, Oct 29, 2024 at 7:35 AM Mark Adams <mfadams at lbl.gov> wrote:
> >>
> >>> Oh my mistake. You are using staggered grids. So you don't have a block
> >>> size that hypre would use for the "nodal" methods.
> >>> I'm not sure what you are doing exactly, but try hypre and you could
> >>> create the null space, zero energy modes, manually, attach to the
> matrix
> >>> and try GAMG.
> >>> You can run with '-info :pc' and grep on GAMG to see if GAMG is picking
> >>> the null space up (send this output if you can't figure it out).
> >>>
> >>> Thanks,
> >>> Mark
> >>>
> >>> On Tue, Oct 29, 2024 at 9:28 AM Mark Adams <mfadams at lbl.gov> wrote:
> >>>
> >>>> This coordinate interface is just a shortcut for vertex based
> >>>> discretizations with 3 dof per vertex, etc. (maybe works in 2D).
> >>>> You will need to construct the null space vectors manually and attach
> it
> >>>> to the matrix. Used by GAMG.
> >>>>
> >>>> Note, for hypre you want to use the "nodal" options and it does not
> use
> >>>> these null space vectors. That is probably the way you want to go.
> >>>> eg: -pc_hypre_boomeramg_nodal_coarsen
> >>>>
> >>>> I would run with hypre boomerang and -help and grep on nodal to see
> all
> >>>> the "nodal" options and use them.
> >>>>
> >>>> Thanks,
> >>>> Mark
> >>>>
> >>>>
> >>>> On Mon, Oct 28, 2024 at 8:06 PM Amneet Bhalla <mail2amneet at gmail.com>
> >>>> wrote:
> >>>>
> >>>>> Hi Folks,
> >>>>>
> >>>>> I am trying to solve the momentum equation in a projection
> >>>>> preconditioner using GAMG or Hypre solver. The equation looks like
> for
> >>>>> velocity variable *v* looks like:
> >>>>>
> >>>>>
> >>>>>  [image: Screenshot 2024-10-28 at 4.15.17 PM.png]
> >>>>>
> >>>>> Here, μ is spatially varying dynamic viscosity and λ is spatially
> >>>>> varying bulk viscosity. I understand that I need to specify rigid
> body
> >>>>> nullspace modes to the multigrid solver in order to accelerate its
> >>>>> convergence. Looking into this routine MatNullSpaceCreateRigidBody()
> (
> >>>>>
> https://urldefense.us/v3/__https://petsc.org/release/manualpages/Mat/MatNullSpaceCreateRigidBody/__;!!G_uCfscf7eWS!bVR6duCoDqPhZrWS-sm1c5qxsFPjZMhdT86AqLpPzWgVy5qoRhd4_Jue2LJOIS6LRrtV2cHGrqger1Yvb-Y5f-0$
> >>>>> <
> https://urldefense.us/v3/__https://petsc.org/release/manualpages/Mat/MatNullSpaceCreateRigidBody/__;!!G_uCfscf7eWS!eKqgIJjCdMzIU76f7X65AmGxrU_-lC7W02BMWafJ77DNf_IuQk6O1X3qU1x9Ez8NJ20vZEL-mF6T1yNmDnwv0eWa2w$
> >),
> >>>>> I see that I need to provide the coordinates of each node. I am using
> >>>>> staggered grid discretization. Do I need to provide coordinates of
> >>>>> staggered grid locations?
> >>>>>
> >>>>> Thanks,
> >>>>> --
> >>>>> --Amneet
> >>>>>
> >>>>>
> >>>>>
> >>>>>
> >>
> >> --
> >> --Amneet
> >>
> >>
> >>
> >>
>


-- 
--Amneet
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