From 8538afe14a36c5bec008c70e0de39c37f2cd5b41 Mon Sep 17 00:00:00 2001 From: bangerth Date: Wed, 21 May 2008 15:04:08 +0000 Subject: [PATCH] Comment on the rapid convergence of the Newton method. git-svn-id: https://svn.dealii.org/trunk@16157 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-33/doc/intro.dox | 5 +- deal.II/examples/step-33/doc/results.dox | 70 +++++++++++++----------- 2 files changed, 42 insertions(+), 33 deletions(-) diff --git a/deal.II/examples/step-33/doc/intro.dox b/deal.II/examples/step-33/doc/intro.dox index f4f6999dbb..1512993e13 100644 --- a/deal.II/examples/step-33/doc/intro.dox +++ b/deal.II/examples/step-33/doc/intro.dox @@ -180,7 +180,10 @@ $\mathbf z$, we arrive at a linear system for $\delta \mathbf W$: @f} This linear system is, in general, neither symmetric nor has any particular definiteness properties. We will either use a direct solver -or Trilinos' GMRES implementation to solve it. +or Trilinos' GMRES implementation to solve it. As will become apparent from +the results shown below, this fully implicit iteration +converges very rapidly (typically in 3 steps) and with the quadratic +convergence order expected from a Newton method.

Auto-Differentiation

diff --git a/deal.II/examples/step-33/doc/results.dox b/deal.II/examples/step-33/doc/results.dox index 458b755113..c56fa2f649 100644 --- a/deal.II/examples/step-33/doc/results.dox +++ b/deal.II/examples/step-33/doc/results.dox @@ -93,41 +93,47 @@ end When we run the program, we get the following kind of output: @code -T=3.14, Number of active cells: 2617 - Number of degrees of freedom: 11184 -NonLin Res: Lin Iter Lin Res -______________________________________ -* 1.684e-02 0007 1.84e-13 -* 4.414e-05 0006 3.82e-15 -* 1.750e-09 0006 1.56e-19 -* 6.831e-16 0000 1.56e-19 -max_df:124 -T=3.16, Number of active cells: 2626 - Number of degrees of freedom: 11220 -NonLin Res: Lin Iter Lin Res -______________________________________ -* 1.673e-02 0007 9.64e-14 -* 4.230e-05 0006 3.83e-15 -* 8.748e-10 0006 7.67e-20 -* 6.804e-16 0000 7.67e-20 -max_df:124 -T=3.18, Number of active cells: 2644 - Number of degrees of freedom: 11296 -NonLin Res: Lin Iter Lin Res -______________________________________ -* 1.674e-02 0007 9.29e-14 -* 3.306e-05 0007 8.07e-17 -* 4.660e-10 0006 4.05e-20 -* 6.898e-16 0000 4.05e-20 -max_df:128 -T=3.2, Number of active cells: 2647 - Number of degrees of freedom: 11312 -NonLin Res: Lin Iter Lin Res -______________________________________ +... +T=0.08 + Number of active cells: 1792 + Number of degrees of freedom: 7656 + + NonLin Res Lin Iter Lin Res + _____________________________________ + 2.424e-02 0008 1.56e-13 + 7.498e-05 0008 1.10e-15 + 8.871e-09 0008 8.58e-20 + 5.998e-16 (converged) + +T=0.1 + Number of active cells: 1798 + Number of degrees of freedom: 7672 + + NonLin Res Lin Iter Lin Res + _____________________________________ + 2.563e-02 0008 1.95e-13 + 9.165e-05 0008 1.10e-15 + 1.234e-08 0008 8.08e-20 + 9.282e-16 (converged) + +T=0.12 + Number of active cells: 1801 + Number of degrees of freedom: 7676 + + NonLin Res Lin Iter Lin Res + _____________________________________ + 2.732e-02 0008 1.72e-13 + 1.147e-04 0008 1.15e-15 + 1.952e-08 0008 1.49e-19 + 1.432e-15 (converged) ... @endcode -This output reports the progress of the Newton iterations and the time stepping. +This output reports the progress of the Newton iterations and the time +stepping. Note that our implementation of the Newton iteration indeed shows +the expected quadratic convergence order: the norm of the nonlinear residual +in each step is roughly the norm of the previous step squared. This leads to +the very rapid convergence we can see here. The result of running these computations is a bunch of output files that we can pass to our visualization program of choice. When we collate them into a -- 2.39.5