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

[Matthew Knepley]

On Mon, Aug 5, 2013 at 8:48 AM, Bishesh Khanal <bisheshkh at gmail.com> wrote:

>
>
>
> 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 ?
>

Yes.


> 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 ?
>

Don't use detect_saddle, but split it by fields -pc_fieldsplit_0_fields
0,1,2 -pc_fieldsplit_1_fields 4

   Matt


>
>>    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
>>
>
>


-- 
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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