From 0b2b72d7246ac56837896f1b120964dc27098a01 Mon Sep 17 00:00:00 2001 From: bangerth Date: Thu, 24 Oct 2013 22:16:11 +0000 Subject: [PATCH] Some minor edits. git-svn-id: https://svn.dealii.org/trunk@31415 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-51/doc/results.dox | 98 +++++++++++++----------- 1 file changed, 53 insertions(+), 45 deletions(-) diff --git a/deal.II/examples/step-51/doc/results.dox b/deal.II/examples/step-51/doc/results.dox index f41ab66065..198cd70344 100644 --- a/deal.II/examples/step-51/doc/results.dox +++ b/deal.II/examples/step-51/doc/results.dox @@ -17,7 +17,13 @@ solution is not continuous on the vertices where the faces meet, even though its values are quite close along lines in the same coordinate direction. The skeleton solution can be interpreted as a rubber spring between the two sides that balances the jumps in the solution (or rather, the flux $\kappa \nabla u -+ \mathbf{c} u$). As the mesh is refined, the jumps between the cells get ++ \mathbf{c} u$). From the picture at the top left, it is clear that +the bulk solution frequently over- and undershoots and that the +skeleton variable in indeed a better approximation to the exact +solution; this explains why we can get a better solution using a +postprocessing step. + +As the mesh is refined, the jumps between the cells get small (we represent a smooth solution), and the skeleton solution approaches the interior parts. For cycle 8, there is no visible difference in the two variables. We also see how boundary conditions are implemented weakly and that @@ -122,16 +128,16 @@ cells dofs val L2 grad L2 val L2-post Q3 elements, global refinement: cells dofs val L2 grad L2 val L2-post - 16 160 3.613e-01 - 1.891e+00 - 3.020e-01 - - 36 336 6.411e-02 4.26 5.081e-01 3.24 3.238e-02 5.51 - 64 576 3.480e-02 2.12 2.533e-01 2.42 5.277e-03 6.31 - 144 1248 8.297e-03 3.54 5.924e-02 3.58 6.330e-04 5.23 - 256 2176 2.254e-03 4.53 1.636e-02 4.47 1.403e-04 5.24 - 576 4800 4.558e-04 3.94 3.277e-03 3.96 1.844e-05 5.01 - 1024 8448 1.471e-04 3.93 1.052e-03 3.95 4.378e-06 5.00 - 2304 18816 2.956e-05 3.96 2.104e-04 3.97 5.750e-07 5.01 - 4096 33280 9.428e-06 3.97 6.697e-05 3.98 1.362e-07 5.01 - 9216 74496 1.876e-06 3.98 1.330e-05 3.99 1.788e-08 5.01 + 16 160 3.613e-01 - 1.891e+00 - 3.020e-01 - + 36 336 6.411e-02 4.26 5.081e-01 3.24 3.238e-02 5.51 + 64 576 3.480e-02 2.12 2.533e-01 2.42 5.277e-03 6.31 + 144 1248 8.297e-03 3.54 5.924e-02 3.58 6.330e-04 5.23 + 256 2176 2.254e-03 4.53 1.636e-02 4.47 1.403e-04 5.24 + 576 4800 4.558e-04 3.94 3.277e-03 3.96 1.844e-05 5.01 + 1024 8448 1.471e-04 3.93 1.052e-03 3.95 4.378e-06 5.00 + 2304 18816 2.956e-05 3.96 2.104e-04 3.97 5.750e-07 5.01 + 4096 33280 9.428e-06 3.97 6.697e-05 3.98 1.362e-07 5.01 + 9216 74496 1.876e-06 3.98 1.330e-05 3.99 1.788e-08 5.01 @endcode @@ -153,41 +159,41 @@ The same convergence rates are observed in 3d. @code Q1 elements, adaptive refinement: cells dofs val L2 grad L2 val L2-post - 8 144 7.122e+00 1.941e+01 6.102e+00 - 29 500 3.309e+00 1.023e+01 2.145e+00 - 113 1792 2.204e+00 1.023e+01 1.912e+00 - 379 5732 6.085e-01 5.008e+00 2.233e-01 - 1317 19412 1.543e-01 1.464e+00 4.196e-02 - 4579 64768 5.058e-02 5.611e-01 9.521e-03 - 14596 199552 2.129e-02 3.122e-01 4.569e-03 - 46180 611400 1.033e-02 1.622e-01 1.684e-03 -144859 1864212 5.007e-03 8.371e-02 7.364e-04 -451060 5684508 2.518e-03 4.562e-02 3.070e-04 + 8 144 7.122e+00 1.941e+01 6.102e+00 + 29 500 3.309e+00 1.023e+01 2.145e+00 + 113 1792 2.204e+00 1.023e+01 1.912e+00 + 379 5732 6.085e-01 5.008e+00 2.233e-01 + 1317 19412 1.543e-01 1.464e+00 4.196e-02 + 4579 64768 5.058e-02 5.611e-01 9.521e-03 + 14596 199552 2.129e-02 3.122e-01 4.569e-03 + 46180 611400 1.033e-02 1.622e-01 1.684e-03 +144859 1864212 5.007e-03 8.371e-02 7.364e-04 +451060 5684508 2.518e-03 4.562e-02 3.070e-04 Q1 elements, global refinement: cells dofs val L2 grad L2 val L2-post - 8 144 7.122e+00 - 1.941e+01 - 6.102e+00 - - 27 432 5.491e+00 0.64 2.184e+01 -0.29 4.448e+00 0.78 - 64 960 3.646e+00 1.42 1.299e+01 1.81 3.306e+00 1.03 - 216 3024 1.595e+00 2.04 8.550e+00 1.03 1.441e+00 2.05 - 512 6912 6.922e-01 2.90 5.306e+00 1.66 2.511e-01 6.07 - 1728 22464 2.915e-01 2.13 2.490e+00 1.87 8.588e-02 2.65 - 4096 52224 1.684e-01 1.91 1.453e+00 1.87 4.055e-02 2.61 - 13824 172800 7.972e-02 1.84 6.861e-01 1.85 1.335e-02 2.74 - 32768 405504 4.637e-02 1.88 3.984e-01 1.89 5.932e-03 2.82 -110592 1354752 2.133e-02 1.92 1.830e-01 1.92 1.851e-03 2.87 + 8 144 7.122e+00 - 1.941e+01 - 6.102e+00 - + 27 432 5.491e+00 0.64 2.184e+01 -0.29 4.448e+00 0.78 + 64 960 3.646e+00 1.42 1.299e+01 1.81 3.306e+00 1.03 + 216 3024 1.595e+00 2.04 8.550e+00 1.03 1.441e+00 2.05 + 512 6912 6.922e-01 2.90 5.306e+00 1.66 2.511e-01 6.07 + 1728 22464 2.915e-01 2.13 2.490e+00 1.87 8.588e-02 2.65 + 4096 52224 1.684e-01 1.91 1.453e+00 1.87 4.055e-02 2.61 + 13824 172800 7.972e-02 1.84 6.861e-01 1.85 1.335e-02 2.74 + 32768 405504 4.637e-02 1.88 3.984e-01 1.89 5.932e-03 2.82 +110592 1354752 2.133e-02 1.92 1.830e-01 1.92 1.851e-03 2.87 Q3 elements, global refinement: cells dofs val L2 grad L2 val L2-post - 8 576 5.670e+00 - 1.868e+01 - 5.462e+00 - - 27 1728 1.048e+00 4.16 6.988e+00 2.42 8.011e-01 4.73 - 64 3840 2.831e-01 4.55 2.710e+00 3.29 1.363e-01 6.16 - 216 12096 7.883e-02 3.15 7.721e-01 3.10 2.158e-02 4.55 - 512 27648 3.642e-02 2.68 3.305e-01 2.95 5.231e-03 4.93 - 1728 89856 8.546e-03 3.58 7.581e-02 3.63 7.640e-04 4.74 - 4096 208896 2.598e-03 4.14 2.313e-02 4.13 1.783e-04 5.06 - 13824 691200 5.314e-04 3.91 4.697e-03 3.93 2.355e-05 4.99 - 32768 1622016 1.723e-04 3.91 1.517e-03 3.93 5.602e-06 4.99 + 8 576 5.670e+00 - 1.868e+01 - 5.462e+00 - + 27 1728 1.048e+00 4.16 6.988e+00 2.42 8.011e-01 4.73 + 64 3840 2.831e-01 4.55 2.710e+00 3.29 1.363e-01 6.16 + 216 12096 7.883e-02 3.15 7.721e-01 3.10 2.158e-02 4.55 + 512 27648 3.642e-02 2.68 3.305e-01 2.95 5.231e-03 4.93 + 1728 89856 8.546e-03 3.58 7.581e-02 3.63 7.640e-04 4.74 + 4096 208896 2.598e-03 4.14 2.313e-02 4.13 1.783e-04 5.06 + 13824 691200 5.314e-04 3.91 4.697e-03 3.93 2.355e-05 4.99 + 32768 1622016 1.723e-04 3.91 1.517e-03 3.93 5.602e-06 4.99 110592 5419008 3.482e-05 3.94 3.055e-04 3.95 7.374e-07 5.00 @endcode @@ -319,7 +325,7 @@ One final note on the efficiency comparison: We tried to use general-purpose sparse matrix structures and similar solvers (optimal AMG preconditioners for both without particular tuning of the AMG parameters on any of them) to give a fair picture of the cost versus accuracy of two methods, on a toy example. It -should be noted however that GMG for continuous finite elements is about a +should be noted however that geometric multigrid (GMG) for continuous finite elements is about a factor four to five faster for p=3 and p=6. The authors of this tutorial have not seen similarly advanced solvers for the HDG linear systems. Also, there are other implementation aspects for CG available such as @@ -327,6 +333,7 @@ fast matrix-free approaches as shown in step-37 that make higher order continuous elements more competitive. Again, it is not clear to the authors of the tutorial whether similar improvements could be made for HDG. +

Possibilities for improvements

As already mentioned in the introduction, one possibility is to implement @@ -398,16 +405,17 @@ components: As can be seen from the table, the solver and assembly calls dominate the -runtime of the program. This also gives a clear indication of where an -improvement makes most sense. +runtime of the program. This also gives a clear indication of where +improvements would make the most sense:
  1. Better linear solvers: We use a BiCGStab iterative solver without preconditioner, where the number of iteration increases with increasing problem size (the number of iterations for Q1 elements and global - refinements start at 35 for the small sizes but increase up to 701 for the + refinements starts at 35 for the small sizes but increase up to 701 for the largest size). To do better, one could for example use an algebraic - multigrid preconditioner from Trilinos. For diffusion-dominated problems as + multigrid preconditioner from Trilinos. For diffusion-dominated + problems such as the problem at hand with finer meshes, such a solver can be designed that uses the matrix-vector products from the more efficient ChunkSparseMatrix on the finest level, as long as we are not working in parallel with MPI. For -- 2.39.5