Hello Salazar,
I would think that the importance of using an optimal
marking strategy such as Dorfler depends on how difficult you think it will
be to see gains with AMR. For example, if your underlying solution does not
have singularities or boundary layers, it becomes more important to try and
equalize errors across elements, make the most of each new dof added and so
on.
However, if the underlying solution does have such sharp features, I would
not expect to see many gains over using an optimal marking strategy over a
more heuristic one like refine_frac or the others which libMesh currently
supports. In such problems, the solution can effectively be seen as a sum
of a singular and non-singular component, and it is the equalization of
error in these two components that we seek first. And AMR should be able to
do this without pretty well needing sophisiticated marking strategies.
The interesting question whether this holds for goal oriented refinement as
well. Note that for non-singular/boundary-layer problems/adjoint systems,
we expect the QoI to converge at twice the rate of the global H1 solution,
with the usual Galerkin (non-stabilized) formulations. Here again,
goal-oriented AMR will only offer an improvement in the convergence
constant, but for an error that is already decreasing fast, and I would
again expect Dorfler type marking strategies to become more important. But
for problems where the forward or adjoint problem has singular behaviour,
we have seen tremendous benefits from AMR using the marking strategies
already present in libMesh.
I am admittedly not well versed with the optimal marking strategies, so if
anyone can pick holes in these arguments, please do so.
Thanks.
On Tue, Feb 14, 2017 at 11:35 AM, Paul T. Bauman <[email protected]> wrote:
> On Mon, Feb 13, 2017 at 6:03 PM, Salazar De Troya, Miguel <
> [email protected]> wrote:
>
> >
> > I’ve read in several papers that in AMR the most common marking strategy
> > is Dorfler marking, which is mathematically grounded.
>
>
> There is some theory that, under some other regularity assumptions (that
> can be proven for quite a few problems), Dorfler marking provides
> optimality for the adaptive refinement process. I admit I'm not deeply
> familiar with the theory. Carsetensen's paper (
> https://arxiv.org/pdf/1312.1171.pdf) and Nochetto's review paper (
> http://www-users.math.umd.edu/~rhn/lectures/adaptivity.pdf) are probably
> good starting points (both have been on my "to read" list for awhile now).
>
>
> > However, I have not seen this implemented in libMesh.
>
>
> It looks as though it's not. Patches welcome! (I would envision this would
> be dropped in MeshRefinement as flag_elements_by_dorfler() or some such.)
>
>
> > I believe that this marking strategy does not consider the coarsening
> > portion of the refinement.
>
>
> I believe there might be some extensions through brief Googling, but I
> haven't read anything in detail. Nevertheless, it is noted that coarsening
> is really only important for unsteady problems as, theoretically, optimal
> adaptive refinement for steady problems does not require coarsening.
>
>
> > On which theories are the current marking strategies libMesh implements
> > based?
> >
>
> Maximum strategies, i.e. refine some fraction of the max (which goes back
> to Babuska).
>
> Best,
>
> Paul
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--
Vikram Garg
Postdoctoral Associate
The University of Texas at Austin
http://vikramvgarg.wordpress.com/
http://www.runforindia.org/runners/vikramg
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