I'll throw out that there's some interesting work being done on advection problems over in the MOOSE framework. You can see this issue in particular which demonstrates use of Discontinuous Galerkin -> Finite Volume to eliminate oscillations (with some really awesome graphics thrown in): https://github.com/idaholab/moose/issues/7967
On 01/17/2017 03:48 PM, Roy Stogner wrote: > On Sun, 15 Jan 2017, Mike Marchywka wrote: > >> http://am2012.math.cas.cz/proceedings/contributions/madden.pdf > This is actually kind of dumbfounding: Fix Your Solutions With One > Weird Trick That Numerical Analysists Hate! I wonder if there's any > extension to 2D/3D? Any way to express the algorithm as a single weak > formulation that gives you a solution function rather than a weird two > phase solve that gives a set of solution points? > > IIRC a lot of FEM stabilization methods history could be (over) > simplified as "(1) see what works for finite differences on uniform > grids, (2) figure out how to copy it in a finite element setting, (3) > figure out how to make it more sensible under some theoretical > criteria, which often makes it better or more generalizable in > practice too". But most of the important step (1) work happened > before I was born, yet this paper (well, the 2007 paper they cite) > sounds like a real fresh entry in that vein. > >> Interested if anyone cares to comment on current status and libmesh >> facilities >> that may be relevant or if this is just a "lmgtfy" case. I gathered the >> normal >> approaches were largely things that modify the apparent diffusivity >> which does not seem to be a lot different from anything else you may add to >> a system. > Most numerical stabilization schemes do boil down into "adding > diffusivity" one way or another, but it's not as hokey as it sounds. > The trick is to do what's called "consistent stabilization": in the > finite element sense, consistent stabilization changes the weak > formulation in such a way that the exact solution to the problem is > still a solution to the stabilized weak problem. This (as well as > making sure the effects are anisotropic in 2D/3D) is generally enough > to get you a good solution (even an exact interpolant, in 1D!) without > smearing out real features. > --- > Roy > > ------------------------------------------------------------------------------ > Check out the vibrant tech community on one of the world's most > engaging tech sites, SlashDot.org! http://sdm.link/slashdot > _______________________________________________ > Libmesh-users mailing list > [email protected] > https://lists.sourceforge.net/lists/listinfo/libmesh-users ------------------------------------------------------------------------------ Check out the vibrant tech community on one of the world's most engaging tech sites, SlashDot.org! http://sdm.link/slashdot _______________________________________________ Libmesh-users mailing list [email protected] https://lists.sourceforge.net/lists/listinfo/libmesh-users
