From 0dac17c6d1664203c8885aa819346f1683067f6d Mon Sep 17 00:00:00 2001 From: Martin Kronbichler Date: Wed, 12 Aug 2009 14:19:40 +0000 Subject: [PATCH] Add my part of the introduction. git-svn-id: https://svn.dealii.org/trunk@19238 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-32/doc/intro.dox | 57 ++++++++++++++++++++++++-- 1 file changed, 54 insertions(+), 3 deletions(-) diff --git a/deal.II/examples/step-32/doc/intro.dox b/deal.II/examples/step-32/doc/intro.dox index 2338c16a18..8e93c15bad 100644 --- a/deal.II/examples/step-32/doc/intro.dox +++ b/deal.II/examples/step-32/doc/intro.dox @@ -37,8 +37,59 @@ which we will discuss first.

Changes to the Stokes preconditioner

-TODO: MARTIN +In this tutorial program, we apply a variant of the preconditioner used in +@ref step_31 "step-31". That preconditioner was built to operate on the +system matrix M in block form such that the product matrix +@f{eqnarray*} + P^{-1} M + = + \left(\begin{array}{cc} + A^{-1} & 0 \\ S^{-1} B A^{-1} & -S^{-1} + \end{array}\right) + \left(\begin{array}{cc} + A & B^T \\ B & 0 + \end{array}\right) +@f} +is of a form that Krylov-based iterative solvers like GMRES can solve in a +few iterations. We then replaced the exact inverse of A by the action +of an AMG preconditioner $\tilde{A}$ based on a vector Laplace matrix, +approximated the Schur complement $S = B A^{-1} B^T$ by a mass matrix $M_p$ +on the pressure space and wrote an InverseMatrix class for +implementing the action of $M_p^{-1}\approx S^{-1}$ on vectors. In the +InverseMatrix class, we used a CG solve with an incomplete Cholesky (IC) +preconditioner for performing the inner solves. +An observation one can make is that we use just the action of a +preconditioner for approximating the velocity inverse $A^{-1}$ (and the +outer GMRES iteration takes care of the approximate character of the +inverse), whereas we use a more or less exact inverse for $M_p^{-1}$, +realized by a fully converged CG solve. What we change here is to skip that +exact inverse matrix and replace it – as usual – by the +action of a preconditioner only. This works, as we will demonstrate +below. For efficiency reasons, we want to avoid increasing the number of +iterations for the block solve. Keep in mind that most of the time in the +solution of the matrix system is the application of the AMG preconditioner +(about half the time of the total solve), and the application of matrix +A (about one third of the total solve time). This means that we +really do not want to do those operations more often when we remove the +inner solve on the Schur complement approximation. It turns out that the +Trilinos IC preconditioner would not fulfill this requirement, however, the +Trilinos ILU preconditioner does. It does even better than so — it +decreases the iteration count for large 3D problems. The reason for that +decrease is that we avoid some errors that CG introduces: Even a converged +solve has still some residual. That is a problem because that small error +interferes with the outer iterative solver, probably because a CG solver +does some nonlinear operations by weighting vectors by some inner products. + +Except the simplification in the preconditioner, we replaced the GMRES +solver by BiCGStab. This is merely to demonstrate that GMRES is not the only +possible option for a saddle point system like the Stokes +equations. BiCGStab harmonizes nicely with the ILU preconditioner on a +pressure mass matrix as approximation for $S^{-1}$, so it is at least as +good as GMRES in this example. Keep in mind the discussion in the results +section of the @ref step_22 "step-22" tutorial program, where we observed +that BiCGStab does not like inner solves with CG, which made us +choose GMRES in step-31

Parallelization on clusters

@@ -199,8 +250,8 @@ and per-cell data. Suffice it to mention that we need the following: BoussinesqFlowProblem::copy_local_to_global_temperature_rhs functions. -We will comment on a few more points in the actual code, but in general their -general structure should be clear from the discussion in @ref threads. +We will comment on a few more points in the actual code, but in general +their structure should be clear from the discussion in @ref threads. The underlying technology for WorkStream identifies "tasks" that need to be worked on (e.g. assembling local contributions on a cell) and schedules these -- 2.39.5