> On Mar 1, 2016, at 12:06 PM, Mohammad Mirzadeh <[email protected]> wrote: > > Nice discussion. > > > On Tue, Mar 1, 2016 at 10:16 AM, Boyce Griffith <[email protected] > <mailto:[email protected]>> wrote: > >> On Mar 1, 2016, at 9:59 AM, Mark Adams <[email protected] >> <mailto:[email protected]>> wrote: >> >> >> >> On Mon, Feb 29, 2016 at 5:42 PM, Boyce Griffith <[email protected] >> <mailto:[email protected]>> wrote: >> >>> On Feb 29, 2016, at 5:36 PM, Mark Adams <[email protected] >>> <mailto:[email protected]>> 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
