<html><head><meta http-equiv="Content-Type" content="text/html charset=us-ascii"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><br class=""><div><blockquote type="cite" class=""><div class="">On Mar 1, 2016, at 12:06 PM, Mohammad Mirzadeh <<a href="mailto:mirzadeh@gmail.com" class="">mirzadeh@gmail.com</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><div dir="ltr" style="font-family: Helvetica; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px;" class="">Nice discussion.<div class=""><br class=""><div class="gmail_extra"><br class=""><div class="gmail_quote">On Tue, Mar 1, 2016 at 10:16 AM, Boyce Griffith<span class="Apple-converted-space"> </span><span dir="ltr" class=""><<a href="mailto:griffith@cims.nyu.edu" target="_blank" class="">griffith@cims.nyu.edu</a>></span><span class="Apple-converted-space"> </span>wrote:<br class=""><blockquote class="gmail_quote" style="margin: 0px 0px 0px 0.8ex; border-left-width: 1px; border-left-color: rgb(204, 204, 204); border-left-style: solid; padding-left: 1ex;"><div style="word-wrap: break-word;" class=""><br class=""><div class=""><span class=""><blockquote type="cite" class=""><div class="">On Mar 1, 2016, at 9:59 AM, Mark Adams <<a href="mailto:mfadams@lbl.gov" target="_blank" class="">mfadams@lbl.gov</a>> wrote:</div><br class=""><div class=""><div dir="ltr" class=""><br class=""><div class="gmail_extra"><br class=""><div class="gmail_quote">On Mon, Feb 29, 2016 at 5:42 PM, Boyce Griffith<span class="Apple-converted-space"> </span><span dir="ltr" class=""><<a href="mailto:griffith@cims.nyu.edu" target="_blank" class="">griffith@cims.nyu.edu</a>></span><span class="Apple-converted-space"> </span>wrote:<br class=""><blockquote class="gmail_quote" style="margin: 0px 0px 0px 0.8ex; border-left-width: 1px; border-left-color: rgb(204, 204, 204); border-left-style: solid; padding-left: 1ex;"><div style="word-wrap: break-word;" class=""><div class=""><div class=""><br class=""><div class=""><blockquote type="cite" class=""><div class="">On Feb 29, 2016, at 5:36 PM, Mark Adams <<a href="mailto:mfadams@lbl.gov" target="_blank" class="">mfadams@lbl.gov</a>> wrote:</div><br class=""><div class=""><div dir="ltr" class=""><div class="gmail_extra"><div class="gmail_quote"><blockquote class="gmail_quote" style="margin: 0px 0px 0px 0.8ex; border-left-width: 1px; border-left-color: rgb(204, 204, 204); border-left-style: solid; padding-left: 1ex;"><div dir="ltr" class=""><div class="gmail_extra"><div class="gmail_quote"><span class=""><blockquote class="gmail_quote" style="margin: 0px 0px 0px 0.8ex; border-left-width: 1px; border-left-color: rgb(204, 204, 204); border-left-style: solid; padding-left: 1ex;"><div dir="ltr" class=""><div class="gmail_extra"><div class="gmail_quote"><div class=""><br class=""></div><div class="">GAMG is use for AMR problems like this a lot in BISICLES.</div></div></div></div></blockquote><div class=""><br class=""></div></span><div class="">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!<span class="Apple-converted-space"> </span></div></div></div></div></blockquote><div class=""><br class=""></div><div class="">No, they are not symmetric. FV is even worse than vertex centered methods. The BCs and the C-F interfaces add non-symmetry.</div></div></div></div></div></blockquote></div><div class=""><br class=""></div></div></div><div class="">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.</div></div></blockquote><div class=""><br class=""></div><div class="">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.</div></div></div></div></div></blockquote><div class=""><br class=""></div></span><div class="">If you are talking about solving Poisson on a composite grid, then refluxing and summing up fluxes are probably the same procedure.</div></div></div></blockquote><div class=""><br class=""></div><div class="">I am not familiar with the terminology used here. What does the refluxing mean?</div><div class=""> </div><blockquote class="gmail_quote" style="margin: 0px 0px 0px 0.8ex; border-left-width: 1px; border-left-color: rgb(204, 204, 204); border-left-style: solid; padding-left: 1ex;"><div style="word-wrap: break-word;" class=""><div class=""><div class=""><br class=""></div><div class="">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 (<a href="http://www.ams.org/journals/mcom/1991-56-194/S0025-5718-1991-1066831-5/S0025-5718-1991-1066831-5.pdf" target="_blank" class="">http://www.ams.org/journals/mcom/1991-56-194/S0025-5718-1991-1066831-5/S0025-5718-1991-1066831-5.pdf</a>) 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.</div><span class=""><font color="#888888" class=""><div class=""><br class=""></div></font></span></div></div></blockquote><div class=""><br class=""></div><div class="">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:<span class="Apple-converted-space"> </span><a href="http://physbam.stanford.edu/~fedkiw/papers/stanford2004-02.pdf" class="">http://physbam.stanford.edu/~fedkiw/papers/stanford2004-02.pdf</a></div><div class=""><br class=""></div><div class="">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. </div></div></div></div></div></div></blockquote><br class=""></div><div>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.</div><div><br class=""></div><div>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.</div><div><div><br class=""></div><div>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.</div><div><br class=""></div><div>-- Boyce</div></div></body></html>