[petsc-users] discontinuous viscosity stokes equation 3D staggered grid

[Bishesh Khanal]

On Mon, Aug 5, 2013 at 3:17 PM, Matthew Knepley <knepley at gmail.com> wrote:

> On Mon, Aug 5, 2013 at 7:54 AM, Bishesh Khanal <bisheshkh at gmail.com>wrote:
>
>>
>>
>>
>> On Wed, Jul 17, 2013 at 9:48 PM, Jed Brown <jedbrown at mcs.anl.gov> wrote:
>>
>>> Bishesh Khanal <bisheshkh at gmail.com> writes:
>>>
>>> > Now, I implemented two different approaches, each for both 2D and 3D,
>>> in
>>> > MATLAB. It works for the smaller sizes but I have problems solving it
>>> for
>>> > the problem size I need (250^3 grid size).
>>> > I use staggered grid with p on cell centers, and components of v on
>>> cell
>>> > faces. Similar split up of K to cell center and faces to account for
>>> the
>>> > variable viscosity case)
>>>
>>> Okay, you're using a staggered-grid finite difference discretization of
>>> variable-viscosity Stokes.  This is a common problem and I recommend
>>> starting with PCFieldSplit with Schur complement reduction (make that
>>> work first, then switch to block preconditioner).  You can use PCLSC or
>>> (probably better for you), assemble a preconditioning matrix containing
>>> the inverse viscosity in the pressure-pressure block.  This diagonal
>>> matrix is a spectrally equivalent (or nearly so, depending on
>>> discretization) approximation of the Schur complement.  The velocity
>>> block can be solved with algebraic multigrid.  Read the PCFieldSplit
>>> docs (follow papers as appropriate) and let us know if you get stuck.
>>>
>>
>> I was trying to assemble the inverse viscosity diagonal matrix to use as
>> the preconditioner for the Schur complement solve step as you suggested.
>> I've few questions about the ways to implement this in Petsc:
>> A naive approach that I can think of would be to create a vector with its
>> components as reciprocal viscosities of the cell centers corresponding to
>> the pressure variables, and then create a diagonal matrix from this vector.
>> However I'm not sure about:
>> How can I make this matrix, (say S_p) compatible to the Petsc
>> distribution of the different rows of the main system matrix over different
>> processors ? The main matrix was created using the DMDA structure with 4
>> dof as explained before.
>> The main matrix correspond to the DMDA with 4 dofs but for the S_p matrix
>> would correspond to only pressure space. Should the distribution of the
>> rows of S_p among different processor not correspond to the distribution of
>> the rhs vector, say h' if it is solving for p with Sp = h' where S = A11
>> inv(A00) A01 ?
>>
>
> PETSc distributed vertices, not dofs, so it never breaks blocks. The P
> distribution is the same as the entire problem divided by 4.
>

Thanks Matt. So if I create a new DMDA with same grid size but with dof=1
instead of 4, the vertices for this new DMDA will be identically
distributed as for the original DMDA ? Or should I inform PETSc by calling
a particular function to make these two DMDA have identical distribution of
the vertices ?

Even then I think there might be a problem due to the presence of
"fictitious pressure vertices". The system matrix (A) contains an identity
corresponding to these fictitious pressure nodes, thus when using a
-pc_fieldsplit_detect_saddle_point, will detect a A11 zero block of size
that correspond to only non-fictitious P-nodes. So the preconditioner S_p
for the Schur complement outer solve with Sp = h' will also need to
correspond to only the non-fictitious P-nodes. This means its size does not
directly correspond to the DMDA grid defined for the original problem.
Could you please suggest an efficient way of assembling this S_p matrix ?


>
>    Matt
>
> --
> 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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