Thank you Jed for the detailed explanation. To tell you the truth, that was a bit overwhelming/scary for an engineer like me-that I have to be supercareful when using those solvers and preconditioners- :D<div>For my case though, I decided to fix the pressure at one node in the domain so I won't end up with a null-space case of the Pressure equation. I assume that way, I would not have to worry about the nullspace problems and the convergence issues you pointed out, right? </div>
<div><br></div><div>Best, </div><div>Mohamad</div><div><br></div><div><br><br><div class="gmail_quote">On Mon, Dec 19, 2011 at 9:09 PM, Jed Brown <span dir="ltr"><<a href="mailto:jedbrown@mcs.anl.gov">jedbrown@mcs.anl.gov</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div class="im"><div class="gmail_quote">On Mon, Dec 19, 2011 at 20:57, Barry Smith <span dir="ltr"><<a href="mailto:bsmith@mcs.anl.gov" target="_blank">bsmith@mcs.anl.gov</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
So please tell use how we SHOULD use AMG with those "indefinite problem produced by most discretizations of incompressible flow" dear teacher :-)</blockquote></div><br></div><div>If only there was a nice complete answer...</div>
<div><br></div><div>We can do block preconditioners advocated by Elman and others. These are the most flexible and the simplest for code reuse. For low Reynolds number, they can also have optimal complexity, although the constants are usually not the best. Most variants are well-supported by PCFieldSplit (e.g. with PCLSC), but some need the user to provide auxiliary operators (e.g. the "pressure convection-diffusion" variant). We could improve support for these cases, but it's a delicate balance and I don't know any way to avoid asking the user to understand a reasonable amount about the method and usually to provide auxiliary information.</div>
<div><br></div><div>We can do coupled multigrid with fieldsplit or "distributed relaxation" as a smoother. These can often be made more robust, but they tend to be more intrusive to implement. These are not usually purely algebraic due to inf-sup issues when coarsening the dual variables (pressure), though Mark Adams' work on this for contact mechanics could be used to coarsen pressure algebraically. I would like to experiment with this in PCGAMG.</div>
<div><br></div><div>We can do coupled multigrid with compatible Vanka-type smoothers. Whether these are algorithmically effective and/or efficient is quite dependent on the discretization. These methods are also usually geometric, though it's possible to algebraically define a Vanka-smoother (though not necessarily efficient). This is straightforward for MAC finite differences on structured grids. For continuous finite elements, the "rotated Q1" Rannacher-Turek elements are most attractive for these smoothers, but Rannacher-Turek elements do not satisfy a discrete Korn's inequality, so they are unusable for many problems. Some variants of DG for incompressible flow seem to be the most interesting for this approach in general domains.</div>
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