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

[Boyce Griffith]

> On Mar 1, 2016, at 2:41 PM, Mohammad Mirzadeh <mirzadeh at gmail.com> wrote:
> 
> 
> 
> On Tue, Mar 1, 2016 at 2:07 PM, Boyce Griffith <griffith at cims.nyu.edu <mailto:griffith at cims.nyu.edu>> wrote:
> 
>> On Mar 1, 2016, at 12:06 PM, Mohammad Mirzadeh <mirzadeh at gmail.com <mailto:mirzadeh at gmail.com>> wrote:
>> 
>> Nice discussion.
>> 
>> 
>> On Tue, Mar 1, 2016 at 10:16 AM, Boyce Griffith <griffith at cims.nyu.edu <mailto:griffith at cims.nyu.edu>> wrote:
>> 
>>> On Mar 1, 2016, at 9:59 AM, Mark Adams <mfadams at lbl.gov <mailto:mfadams at lbl.gov>> wrote:
>>> 
>>> 
>>> 
>>> On Mon, Feb 29, 2016 at 5:42 PM, Boyce Griffith <griffith at cims.nyu.edu <mailto:griffith at cims.nyu.edu>> wrote:
>>> 
>>>> On Feb 29, 2016, at 5:36 PM, Mark Adams <mfadams at lbl.gov <mailto: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 <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 <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.

I don't follow the argument about why it is a bad thing for the fine fluxes to have different values than the overlying coarse flux, but this probably works out to be the same as the Ewing discretization in certain cases (although possibly only in 2D with a refinement ratio of 2).

> 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?

I don't know if I have seen anything that works out all the details for locally refined grids, but LeVeque's extremely readable book, Finite Difference Methods for Ordinary and Partial Differential Equations, works this out for the uniform grid case. If I were doing this on an AMR grid, I would split the domain into a coarse half and a fine half (just one c-f interface) and work out the discrete Green's functions.

-- Boyce
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