[petsc-users] Preconditioner for stokes flow with mixed boundary conditions.

[Abhinav Singh]

The only reference that I know of which solves these equations is :
https://www.pks.mpg.de/fileadmin/user_upload/MPIPKS/group_pages/BiologicalPhysics/juelicher/publications/2015/AHp-MMfIAPVG2015.pdf

There is coordinate free form in the appendix. They have been solved using
the UMFPACK Solver on staggered grids. I am trying a new approach (pressure
correction with auxiliary potential). In 2 dimensions, the approach worked
well with GMRES. In 3d, GMRES again works but with Dirichlet
boundary conditions.

On Mon, 19 Apr 2021 at 20:45, Matthew Knepley <knepley at gmail.com> wrote:

> On Mon, Apr 19, 2021 at 2:38 PM Abhinav Singh <abhinavrajendra at gmail.com>
> wrote:
>
>> Hello,
>>
>> The Stokes flow equations are a 3d version of the equations attached
>> (Stokes-Leslie Flow). Only variables/unknowns are v and
>> u_xy=0.5(Dx(vy)+Dy(vx).
>>
>
> This is not what I have referred to as Stokes flow (Google gave me no
> results for Stokes-Leslie flow). The Stokes operator is elliptic, but I
> have no idea if
> what you have written is. It has fourth order derivatives in it, with
> mixed nonlinear terms, so nothing is clear to me. Is there a
> coordinate-independent form?
> From what I see in the image, I have no idea what solvers might work. Do
> you have any reference where people have solved this before?
>
>   Thanks,
>
>      Matt
>
>
>> I am trying to solve them iteratively by correcting the pressure to reach
>> a steady state. I start with 0 pressure. Currently, I am unable to solve
>> the first iteration (periodic in X and Z. V=0 at Y=0 and Dx(vy)+Dy(vx)=g(x)
>> at Y=10).
>>
>> I think the equations might be singular but I am not sure as in my
>> experience, the problem is well posed if the solution is known at certain
>> boundaries.
>>
>>
>> [image: by default 2021-04-19 at 20.28.28.png]
>>
>> On Mon, 19 Apr 2021 at 15:59, Matthew Knepley <knepley at gmail.com> wrote:
>>
>>> On Mon, Apr 19, 2021 at 9:37 AM Abhinav Singh <abhinavrajendra at gmail.com>
>>> wrote:
>>>
>>>> What does this mean? Stokes means using an incompressibility
>>>>> constraint, for which we often introduce a pressure.
>>>>
>>>> Yes, what I mean is solving the momentum block, say with known
>>>> pressure. viscosity is constant however, the momentum equation has both
>>>> Laplacian and Gradient terms.
>>>>
>>>
>>> That does not make any sense to me. Can you write the equation?
>>>
>>>   Thanks,
>>>
>>>      Matt
>>>
>>>
>>>> You should use a good Laplacian preconditioner, like -pc_type gamg or
>>>>> -pc_type ml.
>>>>
>>>> I tried gamg and it seems to diverge as the solution is NaN. The KSP
>>>> residual message is " 0 KSP Residual norm 1.131782747169e+01  ".
>>>> When using -pc_type ml, I get Aggregate Warning and then some faulty
>>>> address which stops the code.
>>>>
>>>>
>>>>
>>>> On Mon, 19 Apr 2021 at 12:31, Matthew Knepley <knepley at gmail.com>
>>>> wrote:
>>>>
>>>>> On Mon, Apr 19, 2021 at 6:18 AM Abhinav Singh <
>>>>> abhinavrajendra at gmail.com> wrote:
>>>>>
>>>>>> Hello all,
>>>>>>
>>>>>> I am trying to solve for incompressible stokes flow on a particle
>>>>>> based discretization. I use a pressure correction technique along with
>>>>>> Particle strength exchange like operators.
>>>>>>
>>>>>> I call Petsc to solve the Stokes Equation without the pressure term.
>>>>>>
>>>>>
>>>>> What does this mean? Stokes means using an incompressibility
>>>>> constraint, for which we often introduce a pressure.
>>>>>
>>>>> Do you mean you are solving only the momentum block? If so, do you
>>>>> have a constant viscosity? If so, then this is just the Laplace equation.
>>>>> You should use a good Laplacian preconditioner, like -pc_type gamg or
>>>>> -pc_type ml.
>>>>>
>>>>>   Thanks
>>>>>
>>>>>      Matt
>>>>>
>>>>>
>>>>>> GMRES usually works great but with dirichlet boundary conditions.
>>>>>> When I use a mixed boundary condition in Y, (dirichlet on bottom and
>>>>>> Neumann on the top) with periodicity in X,Z. GMRES fails converge when the
>>>>>> size of matrix increases. For smaller size (upto 27*27*5), only GMRES works
>>>>>> and that too only with the option 'pc_type none'. I was unable to find any
>>>>>> preconditioner which worked. Eventually, it also fails for bigger size.
>>>>>> UMFPACK works but LU decomposition fails after a certain size and is very
>>>>>> slow.
>>>>>>
>>>>>> It would be great if you could suggest a way or a preconditioner
>>>>>> which suits this problem.
>>>>>>
>>>>>> Kind regards,
>>>>>> Abhinav
>>>>>>
>>>>>
>>>>>
>>>>> --
>>>>> 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
>>>>>
>>>>> https://www.cse.buffalo.edu/~knepley/
>>>>> <http://www.cse.buffalo.edu/~knepley/>
>>>>>
>>>>
>>>
>>> --
>>> 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
>>>
>>> https://www.cse.buffalo.edu/~knepley/
>>> <http://www.cse.buffalo.edu/~knepley/>
>>>
>>
>
> --
> 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
>
> https://www.cse.buffalo.edu/~knepley/
> <http://www.cse.buffalo.edu/~knepley/>
>
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