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

[Boyce Griffith]

> 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 <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.

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.

-- Boyce
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