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

[Mohammad Mirzadeh]

On Tue, Mar 1, 2016 at 2:07 PM, Boyce Griffith <griffith at cims.nyu.edu>
wrote:

>
> On Mar 1, 2016, at 12:06 PM, Mohammad Mirzadeh <mirzadeh at gmail.com> wrote:
>
> 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.
>
>
> I don't know about that --- check out Table 1 in the paper you cite, which
> seems to indicate first-order convergence in all norms.
>

Sorry my bad. That was the original work which was later extended in
doi:10.1016/j.compfluid.2005.01.006
<http://dx.doi.org/10.1016/j.compfluid.2005.01.006> to second order (c.f.
section 3.3) by using flux weighting in the traverse direction.


>
> The symmetric discretization in the Ewing paper is only slightly more
> complicated, but will give full 2nd-order accuracy in L-1 (and maybe also
> L-2 and L-infinity). One way to think about it is that you are using simple
> linear interpolation at coarse-fine interfaces (3-point interpolation in
> 2D, 4-point interpolation in 3D) using a stencil that is symmetric with
> respect to the center of the coarse grid cell.
>
>
I'll look into that paper. One can never get enough of ideas for C-F
treatments in AMR applications :).


> A (discrete) Green's functions argument explains why one gets higher-order
> convergence despite localized reductions in accuracy along the coarse-fine
> interface --- it has to do with the fact that errors from individual grid
> locations do not have that large of an effect on the solution, and these
> c-f interface errors are concentrated along on a lower dimensional surface
> in the domain.
>

This intuitively makes sense and in fact when you plot the error, you do
see spikes at the C-F interfaces. Do you know of a resource that does a
rigorous analysis of the C-F treatment on the solution error?


>
> -- Boyce
>
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