From 1d74bb495a1202fe488d5145be4fb8720af1835b Mon Sep 17 00:00:00 2001 From: bangerth Date: Fri, 23 Sep 2011 14:42:15 +0000 Subject: [PATCH] Write the solver section. git-svn-id: https://svn.dealii.org/trunk@24398 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-32/doc/intro.dox | 42 +++++++++++++++++++++----- 1 file changed, 34 insertions(+), 8 deletions(-) diff --git a/deal.II/examples/step-32/doc/intro.dox b/deal.II/examples/step-32/doc/intro.dox index f92a65223a..7e94d03c10 100644 --- a/deal.II/examples/step-32/doc/intro.dox +++ b/deal.II/examples/step-32/doc/intro.dox @@ -331,17 +331,43 @@ here that is a bit more complicated (cells are deformed, the pressure varies by orders of magnitude, and we want to plan ahead for more complicated physics), and so we'll change a few things slightly: -- FGMRES instead of GMRES - -- two-stage solver - -- right preconditioner - -- ILU instead of IC +- For more complex problems, it turns out that using just a single AMG V-cycle + as preconditioner is not always sufficient. The outer solver converges just + fine most of the time in a reasonable number of iterations (say, less than + 50) but there are the occasional time step where it suddenly takes 700 or + so. What exactly is going on there is hard to determine, but the problem can + be avoided by using a more accurate solver for the top left + block. Consequently, we'll want to use a CG iteration to invert the top left + block of the preconditioner matrix, and use the AMG as a preconditioner for + the CG solver. + +- The downside of this is that, of course, the Stokes preconditioner becomes + much more expensive (approximately 10 times more expensive than when we just + use a single V-cycle). Our strategy then is this: let's do up to 30 GMRES + iterations with just the V-cycle as a preconditioner and if that doesn't + yield convergence, then take the best approximation of the Stokes solution + obtained after this first round of iterations and use that as the starting + guess for iterations where we use the full inner solver with a rather + lenient tolerance as preconditioner. In all our experiments this leads to + convergence in only a few additional iterations. + +- One thing we need to pay attention to is that when using a CG with a lenient + tolerance in the preconditioner, then $y = \tilde A^{-1} r$ is no longer a + linear function of $r$ (it is, of course, if we have a very stringent + tolerance in our solver, or if we only apply a single V-cycle). This is a + problem since now our preconditioner is no longer a linear operator; in + other words, every time GMRES uses it the preconditioner looks + different. The standard GMRES solver can't deal with this, leading to slow + convergence or even breakdown, but the F-GMRES variant is designed to deal + with exactly this kind of situation and we consequently use it. + +@todo Couldn't we use GMRES for the first stage solve and F-GMRES for the +second stage only. Does it make a difference? Is F-GMRES slower? @todo Why again did we use a right preconditioner when in step-31 we use a -left preconditioner? +left preconditioner? or do we? +@todo Why do we use an ILU instead of an IC for S as in step-31? As a final note, let us remark that in step-31 we computed the Schur complement $S=B A^{-1} B^T$ by approximating -- 2.39.5