I'm working on some adaptive discontinuous Galerkin codes for time harmonic wave propagation, currently just Helmholtz, but will be branching out once I have a working prototype in this case.

There are some papers out there demonstrating that residual based a posteriori estimators combined with a proper marking strategy yield a provable error reduction provided the refinement satisfies some conditions.

(e.g. http://imajna.oxfordjournals.org/content/early/2012/11/23/imanum.drs028.abstract )

Now I am not an expert on a posteriori estimation. However the conditions of these theorems for error reduction appear to rely strongly on the fact that the solutions satisfy the specified discrete weak forms exactly. Unfortunately a direct solve is only possible for me on a few levels of refinement, then the problem becomes quite large. (The initial mesh has to be fine enough to resolve the wave frequency to ensure the estimators are valid, so it's not necessarily the case that I'm beginning with only a few elements)

I can't however find any literature detailing the relationship between the error incurred in an inexact linear solve and its impact on the reliability of convergence theorems for adaptivity. Is there any out there I might be missing, or maybe I'm just asking for too much?

  • $\begingroup$ There are some recent papers by Dirk Praetorius' group at TU Wien. $\endgroup$ – timur Dec 24 '13 at 20:52

There is a paper from the 1990s (early 2000s) by Roland Becker on the topic. I think it was co-authored by Rolf Rannacher.

There is also a more recent paper by Rannacher and Vihharev that I proof read last year. It may not have appeared, but you could ask them whether they're willing to send you a copy. It is not this one http://numerik.iwr.uni-heidelberg.de/Paper/MeidnerRannacherVihharev_paper.pdf but that one may also be of interest to you.

  • $\begingroup$ Thanks, this was helpful. I'm still curious about the quasi-orthogonality results which form the bedrock of convergence theorems though. I imagine this would be a very difficult thing to prove with your standard iterative technique, since the residuals are often measured in a totally unrelated norm to the energy norm employed from finite element methods. But any result in this direction would be helpful. $\endgroup$ – Reid.Atcheson Jul 19 '13 at 2:48

You might get some idea from Stevenson's original paper published in FOCM, because while there you have a continuous Galerkin method, the analysis is done with inexact solvers.

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    $\begingroup$ Welcome to Scicomp.SE! Your answer would be even more helpful if you could give a full reference (or even a link) to the paper you refer to. $\endgroup$ – Christian Clason Nov 29 '13 at 7:45

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