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

[Boyce Griffith]

> 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 <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.

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.

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.

-- Boyce
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