Adjust output, mention change

git-svn-id: https://svn.dealii.org/trunk@30451 0785d39b-7218-0410-832d-ea1e28bc413d

diff --git a/deal.II/doc/news/changes.h b/deal.II/doc/news/changes.h
index 6f9e1513db654a106308c0234c42dfd9ace49249..95099c18b9bbb24a7210e3902b5fda88f110b1df 100644 (file)
--- a/deal.II/doc/news/changes.h
+++ b/deal.II/doc/news/changes.h
@@ -68,6 +68,16 @@ inconvenience this causes.
 <h3>Specific improvements</h3>
 
 <ol>
+  <li>
+  Improved: MappingQ now uses the points of the Gauss-Lobatto quadrature
+  formula as support points instead of equispaced ones. This allows its use
+  for high polynomial orders and also gives better interpolation of circular
+  boundaries. Beware that mappings of order three and higher will behave
+  slightly differently now (usually better).
+  <br>
+  (Martin Kronbichler, 2013/08/23)
+  </li>
+
   <li>
   Improved: Several .cc files in the deal.II directory have been split in
   order to better utilize multiple processors when compiling in parallel and

diff --git a/deal.II/examples/step-10/doc/results.dox b/deal.II/examples/step-10/doc/results.dox
index 2e25ba40fad1ccede1cb838ace3de3dc58a9c29e..dcc5dac01145ceb900348585861700a1f5b74ae8 100644 (file)
--- a/deal.II/examples/step-10/doc/results.dox
+++ b/deal.II/examples/step-10/doc/results.dox
@@ -9,7 +9,7 @@ the commands
 @code
 set style data lines
 set size 0.721, 1
-set nokey
+unset key
 plot [-1:1][-1:1] "ball0_mapping_q1.dat"
 @endcode
 or using one of the other filenames. The second line makes sure that
@@ -90,104 +90,101 @@ good accuracy. This is the output of this part of the program:
 Computation of Pi by the area:
 ==============================
 Degree = 1
-cells         eval.pi         error                           
-5       1.9999999999999998      1.1416e+00      -
-20      2.8284271247461898      3.1317e-01      1.87
-80      3.0614674589207178      8.0125e-02      1.97
-320     3.1214451522580520      2.0148e-02      1.99
-1280    3.1365484905459389      5.0442e-03      2.00
-5120    3.1403311569547521      1.2615e-03      2.00
+cells      eval.pi            error      
+    5 1.9999999999999993 1.1416e+00    - 
+   20 2.8284271247461894 3.1317e-01 1.87 
+   80 3.0614674589207178 8.0125e-02 1.97 
+  320 3.1214451522580520 2.0148e-02 1.99 
+ 1280 3.1365484905459393 5.0442e-03 2.00 
+ 5120 3.1403311569547534 1.2615e-03 2.00 
 
 Degree = 2
-cells         eval.pi         error                           
-5       3.1045694996615869      3.7023e-02      -
-20      3.1391475703122276      2.4451e-03      3.92
-80      3.1414377167038303      1.5494e-04      3.98
-320     3.1415829366419019      9.7169e-06      4.00
-1280    3.1415920457576907      6.0783e-07      4.00
-5120    3.1415926155921126      3.7998e-08      4.00
+cells      eval.pi            error      
+    5 3.1045694996615865 3.7023e-02    - 
+   20 3.1391475703122271 2.4451e-03 3.92 
+   80 3.1414377167038303 1.5494e-04 3.98 
+  320 3.1415829366419015 9.7169e-06 4.00 
+ 1280 3.1415920457576911 6.0783e-07 4.00 
+ 5120 3.1415926155921139 3.7998e-08 4.00 
 
 Degree = 3
-cells         eval.pi         error                           
-5       3.1465390309173475      4.9464e-03      -
-20      3.1419461263297386      3.5347e-04      3.81
-80      3.1416154689089382      2.2815e-05      3.95
-320     3.1415940909713274      1.4374e-06      3.99
-1280    3.1415927436051230      9.0015e-08      4.00
-5120    3.1415926592185492      5.6288e-09      4.00
+cells      eval.pi            error      
+    5 3.1410033851499310 5.8927e-04    - 
+   20 3.1415830393583861 9.6142e-06 5.94 
+   80 3.1415925017363837 1.5185e-07 5.98 
+  320 3.1415926512106722 2.3791e-09 6.00 
+ 1280 3.1415926535525962 3.7197e-11 6.00 
+ 5120 3.1415926535892140 5.7923e-13 6.00 
 
 Degree = 4
-cells         eval.pi         error                           
-5       3.1418185737113964      2.2592e-04      -
-20      3.1415963919525050      3.7384e-06      5.92
-80      3.1415927128397780      5.9250e-08      5.98
-320     3.1415926545188264      9.2903e-10      5.99
-1280    3.1415926536042722      1.4479e-11      6.00
-5120    3.1415926535899668      1.7343e-13      6.38
-
+cells      eval.pi            error       
+    5 3.1415871927401127 5.4608e-06     - 
+   20 3.1415926314742437 2.2116e-08  7.95 
+   80 3.1415926535026228 8.7170e-11  7.99 
+  320 3.1415926535894529 3.4036e-13  8.00 
+ 1280 3.1415926535897927 2.9187e-16 10.19 
+ 5120 3.1415926535897944 1.3509e-15 -2.21 
 
 Computation of Pi by the perimeter:
 ===================================
 Degree = 1
-cells         eval.pi         error                           
-5       2.8284271247461903      3.1317e-01      -
-20      3.0614674589207183      8.0125e-02      1.97
-80      3.1214451522580524      2.0148e-02      1.99
-320     3.1365484905459393      5.0442e-03      2.00
-1280    3.1403311569547525      1.2615e-03      2.00
-5120    3.1412772509327729      3.1540e-04      2.00
+cells      eval.pi            error      
+    5 2.8284271247461898 3.1317e-01    - 
+   20 3.0614674589207178 8.0125e-02 1.97 
+   80 3.1214451522580520 2.0148e-02 1.99 
+  320 3.1365484905459393 5.0442e-03 2.00 
+ 1280 3.1403311569547525 1.2615e-03 2.00 
+ 5120 3.1412772509327729 3.1540e-04 2.00 
 
 Degree = 2
-cells         eval.pi         error                           
-5       3.1248930668550599      1.6700e-02      -
-20      3.1404050605605454      1.1876e-03      3.81
-80      3.1415157631807014      7.6890e-05      3.95
-320     3.1415878042798613      4.8493e-06      3.99
-1280    3.1415923498174538      3.0377e-07      4.00
-5120    3.1415926345932004      1.8997e-08      4.00
+cells      eval.pi            error      
+    5 3.1248930668550594 1.6700e-02    - 
+   20 3.1404050605605449 1.1876e-03 3.81 
+   80 3.1415157631807014 7.6890e-05 3.95 
+  320 3.1415878042798617 4.8493e-06 3.99 
+ 1280 3.1415923498174534 3.0377e-07 4.00 
+ 5120 3.1415926345932004 1.8997e-08 4.00 
 
 Degree = 3
-cells         eval.pi         error                           
-5       3.1442603311164286      2.6677e-03      -
-20      3.1417729561193588      1.8030e-04      3.89
-80      3.1416041192612365      1.1466e-05      3.98
-320     3.1415933731961760      7.1961e-07      3.99
-1280    3.1415926986118001      4.5022e-08      4.00
-5120    3.1415926564043946      2.8146e-09      4.00
+cells      eval.pi            error      
+    5 3.1414940401456057 9.8613e-05    - 
+   20 3.1415913432549156 1.3103e-06 6.23 
+   80 3.1415926341726914 1.9417e-08 6.08 
+  320 3.1415926532906893 2.9910e-10 6.02 
+ 1280 3.1415926535851360 4.6571e-12 6.01 
+ 5120 3.1415926535897203 7.2845e-14 6.00 
 
 Degree = 4
-cells         eval.pi         error                           
-5       3.1417078926581086      1.1524e-04      -
-20      3.1415945317216001      1.8781e-06      5.94
-80      3.1415926832497720      2.9660e-08      5.98
-320     3.1415926540544636      4.6467e-10      6.00
-1280    3.1415926535970535      7.2602e-12      6.00
-5120    3.1415926535899010      1.0805e-13      6.07
+cells      eval.pi            error      
+    5 3.1415921029432576 5.5065e-07    - 
+   20 3.1415926513737600 2.2160e-09 7.96 
+   80 3.1415926535810712 8.7218e-12 7.99 
+  320 3.1415926535897594 3.3668e-14 8.02 
+ 1280 3.1415926535897922 1.0617e-15 4.99 
+ 5120 3.1415926535897931 1.0061e-16 3.40 
 @endcode
 
 
 
-One of the immediate observations from the output is that in all cases
-the values converge quickly to the true value of
-$\pi=3.141592653589793238462643$. Note that for the $Q_4$ mapping, the last
-number is correct to 13 digits in both computations, which is already
-quite a lot. However, also note that for the $Q_1$ mapping, even on the
-finest grid the accuracy is significantly worse than on the coarse
-grid for a $Q_4$ mapping!
+One of the immediate observations from the output is that in all cases the
+values converge quickly to the true value of
+$\pi=3.141592653589793238462643$. Note that for the $Q_4$ mapping, we are
+already in the regime of roundoff errors and the convergence rate levels off,
+which is already quite a lot. However, also note that for the $Q_1$ mapping,
+even on the finest grid the accuracy is significantly worse than on the coarse
+grid for a $Q_3$ mapping!
 
 
 
-The last column of the output shows the convergence order, in powers
-of the mesh width $h$. In the introduction, we had stated that
-the convergence order for a $Q_p$ mapping should be
-$h^{p+1}$. However, in the example shown, the $Q_2$ and $Q_4$
-mappings show a convergence order of $h^{p+2}$! This at
-first surprising fact is readily explained by the particular boundary
-we have chosen in this example. In fact, the circle is described by the function
-$\sqrt{1-x^2}$, which has the series expansion 
-$1-x^2/2-x^4/8-x^6/16+\ldots$
-around $x=0$. Thus, for the quadratic mapping where the
-truncation error of the quadratic approximation should be cubic, there
-is no such term but only a quartic one, which raises the convergence
-order to 4, instead of 3. The same happens for the $Q_4$ mapping.
+The last column of the output shows the convergence order, in powers of the
+mesh width $h$. In the introduction, we had stated that the convergence order
+for a $Q_p$ mapping should be $h^{p+1}$. However, in the example shown, the
+order is rather $h^{2p}$! This at first surprising fact is explained by the
+properties of the $Q_p$ mapping. At order <i>p</i>, it uses support points
+that are based on the <i>p</i>+1 point Gauss-Lobatto quadrature rule that
+selects the support points in such a way that the quadrature rule converges at
+order 2<i>p</i>. Even though these points are here only used for interpolation
+of a <i>p</i>th order polynomial, we get a superconvergence effect when
+numerically evaluating the integral that actually gives this high order of
+convergence.