[petsc-users] Neumann BC with non-symmetric matrix

[Mohammad Mirzadeh]

Nice discussion.


On Tue, Mar 1, 2016 at 10:16 AM, Boyce Griffith <griffith at cims.nyu.edu>
wrote:

>
> On Mar 1, 2016, at 9:59 AM, Mark Adams <mfadams at lbl.gov> wrote:
>
>
>
> On Mon, Feb 29, 2016 at 5:42 PM, Boyce Griffith <griffith at cims.nyu.edu>
> wrote:
>
>>
>> On Feb 29, 2016, at 5:36 PM, Mark Adams <mfadams at lbl.gov> wrote:
>>
>>
>>>> GAMG is use for AMR problems like this a lot in BISICLES.
>>>>
>>>
>>> Thanks for the reference. However, a quick look at their paper suggests
>>> they are using a finite volume discretization which should be symmetric and
>>> avoid all the shenanigans I'm going through!
>>>
>>
>> No, they are not symmetric.  FV is even worse than vertex centered
>> methods.  The BCs and the C-F interfaces add non-symmetry.
>>
>>
>> If you use a different discretization, it is possible to make the c-f
>> interface discretization symmetric --- but symmetry appears to come at a
>> cost of the reduction in the formal order of accuracy in the flux along the
>> c-f interface. I can probably dig up some code that would make it easy to
>> compare.
>>
>
> I don't know.  Chombo/Boxlib have a stencil for C-F and do F-C with
> refluxing, which I do not linearize.  PETSc sums fluxes at faces directly,
> perhaps this IS symmetric? Toby might know.
>
>
> If you are talking about solving Poisson on a composite grid, then
> refluxing and summing up fluxes are probably the same procedure.
>

I am not familiar with the terminology used here. What does the refluxing
mean?


>
> Users of these kinds of discretizations usually want to use the
> conservative divergence at coarse-fine interfaces, and so the main question
> is how to set up the viscous/diffusive flux stencil at coarse-fine
> interfaces (or, equivalently, the stencil for evaluating ghost cell values
> at coarse-fine interfaces). It is possible to make the overall
> discretization symmetric if you use a particular stencil for the flux
> computation. I think this paper (
> http://www.ams.org/journals/mcom/1991-56-194/S0025-5718-1991-1066831-5/S0025-5718-1991-1066831-5.pdf)
> is one place to look. (This stuff is related to "mimetic finite difference"
> discretizations of Poisson.) This coarse-fine interface discretization
> winds up being symmetric (although possibly only w.r.t. a weighted inner
> product --- I can't remember the details), but the fluxes are only
> first-order accurate at coarse-fine interfaces.
>
>
Right. I think if the discretization is conservative, i.e. discretizing div
of grad, and is compact, i.e. only involves neighboring cells sharing a
common face, then it is possible to construct symmetric discretization. An
example, that I have used before in other contexts, is described here:
http://physbam.stanford.edu/~fedkiw/papers/stanford2004-02.pdf

An interesting observation is although the fluxes are only first order
accurate, the final solution to the linear system exhibits super
convergence, i.e. second-order accurate, even in L_inf. Similar behavior is
observed with non-conservative, node-based finite difference
discretizations.

-- Boyce
>
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