--- /dev/null
+// ---------------------------------------------------------------------
+// $Id: integrate_one_over_r.cc 30338 2013-08-18 22:02:27Z heltai $
+//
+// Copyright (C) 2005 - 2013 by the deal.II authors
+//
+// This file is part of the deal.II library.
+//
+// The deal.II library is free software; you can use it, redistribute
+// it, and/or modify it under the terms of the GNU Lesser General
+// Public License as published by the Free Software Foundation; either
+// version 2.1 of the License, or (at your option) any later version.
+// The full text of the license can be found in the file LICENSE at
+// the top level of the deal.II distribution.
+//
+// ---------------------------------------------------------------------
+
+// integrates the function *f(x,y)/R, where f(x,y) is a power of x and
+// y on the set [0,1]x[0,1]. dim = 2 only.
+
+#include <fstream>
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/utilities.h>
+
+// all include files needed for the program
+#include <deal.II/base/quadrature_lib.h>
+#include <deal.II/base/geometry_info.h>
+#include <deal.II/fe/fe_q.h>
+
+#include <fstream>
+#include <string>
+
+#include <math.h>
+
+using namespace std;
+using namespace dealii;
+
+// We test the integration of singular kernels with a singularity of kind 1/R
+// We multiply this function with a polynomial up to degree 6.
+
+double
+exact_integral_one_over_r (
+ const unsigned int i, const unsigned int j,
+ const unsigned int vertex_index);
+
+ofstream logfile("output");
+
+int
+main ()
+{
+ deallog.attach(logfile);
+ deallog.depth_console(0);
+ deallog << std::fixed;
+
+ deallog << endl
+ << "Calculation of the integral of f(x,y)*1/R on [0,1]x[0,1]" << endl
+ << "for f(x,y) = x^i y^j, with i,j ranging from 0 to 5, and R being" << endl
+ << "the distance from (x,y) to four vertices of the square." << endl
+ << endl;
+
+
+ std::vector<Point<2> > vertices = FE_Q<2>(1).get_unit_support_points();
+
+ for (unsigned int m=1; m<7; ++m)
+ {
+ deallog << " =========Quadrature Order: " << m
+ << " =============================== " << endl;
+ deallog << " ============================================================ " << endl;
+ unsigned int index=0;
+ {
+ deallog << " ===============Vertex: " << vertices[index]
+ << " ============================= " << endl;
+ QTelles<2> quad(m, vertices[index]);
+ QGaussOneOverR<2> quad2(m, vertices[index]);
+
+
+ for (unsigned int i = 0; i < 6; ++i)
+ {
+ for (unsigned int j = 0; j < 6; ++j)
+ {
+
+ double exact_integral = exact_integral_one_over_r(index, i,j);
+ double approx_integral = 0;
+ double approx_integral_2 = 0;
+
+ for (unsigned int q=0; q< quad.size(); ++q)
+ {
+ double x = quad.point(q)[0];
+ double y = quad.point(q)[1];
+ double R = sqrt(x*x+y*y);
+ approx_integral += ( pow(x, (double)i) *
+ pow(y, (double)j) / R *
+ quad.weight(q) );
+ }
+
+ for (unsigned int q=0; q< quad2.size(); ++q)
+ {
+ double x = quad2.point(q)[0];
+ double y = quad2.point(q)[1];
+ double R = sqrt(x*x+y*y);
+ approx_integral_2 += ( pow(x, (double)i) *
+ pow(y, (double)j) *
+ quad2.weight(q) );
+ }
+
+ deallog << "f(x,y) = x^" << i
+ << " y^" << j
+ << ", Errors = "
+ << approx_integral - exact_integral
+ << ", "
+ << approx_integral_2 - exact_integral
+ << std::endl;
+ }
+ }
+ }
+ }
+ }
+
+double exact_integral_one_over_r(const unsigned int vertex_index,
+ const unsigned int i,
+ const unsigned int j)
+ {
+ Assert(vertex_index < 4, ExcInternalError());
+ Assert(i<6, ExcNotImplemented());
+ Assert(j<6, ExcNotImplemented());
+
+// The integrals are computed using the following maple snippet of
+// code:
+//
+// singint := proc(index, N, M)
+// if index = 0 then
+// return int(int(x^N *y^M/sqrt(x^2+y^2), x=0.0..1.0), y=0.0..1.0);
+// elif index = 1 then
+// return int(int(x^N *y^M/sqrt((x-1)^2+y^2), x=0.0..1.0), y=0.0..1.0);
+// elif index = 2 then
+// return int(int(x^N *y^M/sqrt(x^2+(y-1)^2), x=0.0..1.0), y=0.0..1.0);
+// elif index = 3 then
+// return int(int((1-x)^N *(1-y)^M/sqrt(x^2+y^2), x=0.0..1.0), y=0.0..1.0);
+// end if;
+// end proc;
+// Digits := 20;
+// for i from 3 to 3 do
+// for n from 0 to 5 do
+// for m from 0 to 5 do
+// v[i+1][n+1][m+1] = sing_int(i, n, m);
+// end do;
+// end do;
+// end do;
+// C(v)
+
+ static double v[4][6][6] =
+ {
+ {
+ { 0}}};
+ if (v[0][0][0] == 0)
+ {
+ v[0][0][0] = 0.17627471740390860505e1;
+ v[0][0][1] = 0.64779357469631903702e0;
+ v[0][0][2] = 0.38259785823210634567e0;
+ v[0][0][3] = 0.26915893322379450224e0;
+ v[0][0][4] = 0.20702239737104695572e0;
+ v[0][0][5] = 0.16800109713227567467e0;
+ v[0][1][0] = 0.64779357469631903702e0;
+ v[0][1][1] = 0.27614237491539669920e0;
+ v[0][1][2] = 0.17015838751246776515e0;
+ v[0][1][3] = 0.12189514164974600651e0;
+ v[0][1][4] = 0.94658660368131133694e-1;
+ v[0][1][5] = 0.77263794021029438797e-1;
+ v[0][2][0] = 0.38259785823210634567e0;
+ v[0][2][1] = 0.17015838751246776515e0;
+ v[0][2][2] = 0.10656799507071040471e0;
+ v[0][2][3] = 0.76947022258735165920e-1;
+ v[0][2][4] = 0.60022626787495395021e-1;
+ v[0][2][5] = 0.49131622931360879320e-1;
+ v[0][3][0] = 0.26915893322379450224e0;
+ v[0][3][1] = 0.12189514164974600651e0;
+ v[0][3][2] = 0.76947022258735165919e-1;
+ v[0][3][3] = 0.55789184535895709637e-1;
+ v[0][3][4] = 0.43625068213915842136e-1;
+ v[0][3][5] = 0.35766126849971778500e-1;
+ v[0][4][0] = 0.20702239737104695572e0;
+ v[0][4][1] = 0.94658660368131133694e-1;
+ v[0][4][2] = 0.60022626787495395021e-1;
+ v[0][4][3] = 0.43625068213915842137e-1;
+ v[0][4][4] = 0.34164088852375945192e-1;
+ v[0][4][5] = 0.28037139560980277614e-1;
+ v[0][5][0] = 0.16800109713227567467e0;
+ v[0][5][1] = 0.77263794021029438797e-1;
+ v[0][5][2] = 0.49131622931360879320e-1;
+ v[0][5][3] = 0.35766126849971778501e-1;
+ v[0][5][4] = 0.28037139560980277614e-1;
+ v[0][5][5] = 0.23024181049838367777e-1;
+ v[1][0][0] = 0.17627471740390860505e1;
+ v[1][0][1] = 0.64779357469631903702e0;
+ v[1][0][2] = 0.38259785823210634567e0;
+ v[1][0][3] = 0.26915893322379450224e0;
+ v[1][0][4] = 0.20702239737104695572e0;
+ v[1][0][5] = 0.16800109713227567467e0;
+ v[1][1][0] = 0.11149535993427670134e1;
+ v[1][1][1] = 0.37165119978092233782e0;
+ v[1][1][2] = 0.21243947071963858053e0;
+ v[1][1][3] = 0.14726379157404849573e0;
+ v[1][1][4] = 0.11236373700291582202e0;
+ v[1][1][5] = 0.90737303111246235871e-1;
+ v[1][2][0] = 0.84975788287855432210e0;
+ v[1][2][1] = 0.26566721237799340376e0;
+ v[1][2][2] = 0.14884907827788122009e0;
+ v[1][2][3] = 0.10231567218303765515e0;
+ v[1][2][4] = 0.77727703422280083352e-1;
+ v[1][2][5] = 0.62605132021577676395e-1;
+ v[1][3][0] = 0.69800109142265347423e0;
+ v[1][3][1] = 0.20794647083778622837e0;
+ v[1][3][2] = 0.11487965864809909847e0;
+ v[1][3][3] = 0.78525390514866270852e-1;
+ v[1][3][4] = 0.59489228415223897572e-1;
+ v[1][3][5] = 0.47838457013298217744e-1;
+ v[1][4][0] = 0.59754668912231692323e0;
+ v[1][4][1] = 0.17125249387868593878e0;
+ v[1][4][2] = 0.93606816359052444729e-1;
+ v[1][4][3] = 0.63728830247554475330e-1;
+ v[1][4][4] = 0.48187332620207367724e-1;
+ v[1][4][5] = 0.38708290797416359020e-1;
+ v[1][5][0] = 0.52527944036356840363e0;
+ v[1][5][1] = 0.14574366656617935708e0;
+ v[1][5][2] = 0.78997159795636003667e-1;
+ v[1][5][3] = 0.53620816423066464705e-1;
+ v[1][5][4] = 0.40487985967086264433e-1;
+ v[1][5][5] = 0.32498604596082509165e-1;
+ v[2][0][0] = 0.17627471740390860505e1;
+ v[2][0][1] = 0.11149535993427670134e1;
+ v[2][0][2] = 0.84975788287855432210e0;
+ v[2][0][3] = 0.69800109142265347419e0;
+ v[2][0][4] = 0.59754668912231692318e0;
+ v[2][0][5] = 0.52527944036356840362e0;
+ v[2][1][0] = 0.64779357469631903702e0;
+ v[2][1][1] = 0.37165119978092233782e0;
+ v[2][1][2] = 0.26566721237799340376e0;
+ v[2][1][3] = 0.20794647083778622835e0;
+ v[2][1][4] = 0.17125249387868593876e0;
+ v[2][1][5] = 0.14574366656617935708e0;
+ v[2][2][0] = 0.38259785823210634567e0;
+ v[2][2][1] = 0.21243947071963858053e0;
+ v[2][2][2] = 0.14884907827788122009e0;
+ v[2][2][3] = 0.11487965864809909845e0;
+ v[2][2][4] = 0.93606816359052444712e-1;
+ v[2][2][5] = 0.78997159795636003667e-1;
+ v[2][3][0] = 0.26915893322379450223e0;
+ v[2][3][1] = 0.14726379157404849572e0;
+ v[2][3][2] = 0.10231567218303765514e0;
+ v[2][3][3] = 0.78525390514866270835e-1;
+ v[2][3][4] = 0.63728830247554475311e-1;
+ v[2][3][5] = 0.53620816423066464702e-1;
+ v[2][4][0] = 0.20702239737104695572e0;
+ v[2][4][1] = 0.11236373700291582202e0;
+ v[2][4][2] = 0.77727703422280083352e-1;
+ v[2][4][3] = 0.59489228415223897563e-1;
+ v[2][4][4] = 0.48187332620207367713e-1;
+ v[2][4][5] = 0.40487985967086264434e-1;
+ v[2][5][0] = 0.16800109713227567468e0;
+ v[2][5][1] = 0.90737303111246235879e-1;
+ v[2][5][2] = 0.62605132021577676399e-1;
+ v[2][5][3] = 0.47838457013298217740e-1;
+ v[2][5][4] = 0.38708290797416359014e-1;
+ v[2][5][5] = 0.32498604596082509169e-1;
+ v[3][0][0] = 0.17627471740390860505e1;
+ v[3][0][1] = 0.11149535993427670134e1;
+ v[3][0][2] = 0.84975788287855432210e0;
+ v[3][0][3] = 0.69800109142265347419e0;
+ v[3][0][4] = 0.59754668912231692318e0;
+ v[3][0][5] = 0.52527944036356840362e0;
+ v[3][1][0] = 0.11149535993427670134e1;
+ v[3][1][1] = 0.74330239956184467563e0;
+ v[3][1][2] = 0.58409067050056091834e0;
+ v[3][1][3] = 0.49005462058486724584e0;
+ v[3][1][4] = 0.42629419524363098443e0;
+ v[3][1][5] = 0.37953577379738904654e0;
+ v[3][2][0] = 0.84975788287855432210e0;
+ v[3][2][1] = 0.58409067050056091834e0;
+ v[3][2][2] = 0.46727253640044873467e0;
+ v[3][2][3] = 0.39698780839518011595e0;
+ v[3][2][4] = 0.34864851772399749038e0;
+ v[3][2][5] = 0.31278926702684569312e0;
+ v[3][3][0] = 0.69800109142265347423e0;
+ v[3][3][1] = 0.49005462058486724586e0;
+ v[3][3][2] = 0.39698780839518011599e0;
+ v[3][3][3] = 0.34027526433872581371e0;
+ v[3][3][4] = 0.30088082631586196583e0;
+ v[3][3][5] = 0.27141910362887187844e0;
+ v[3][4][0] = 0.59754668912231692323e0;
+ v[3][4][1] = 0.42629419524363098445e0;
+ v[3][4][2] = 0.34864851772399749044e0;
+ v[3][4][3] = 0.30088082631586196576e0;
+ v[3][4][4] = 0.26744962339187730308e0;
+ v[3][4][5] = 0.24229245314748740295e0;
+ v[3][5][0] = 0.52527944036356840363e0;
+ v[3][5][1] = 0.37953577379738904655e0;
+ v[3][5][2] = 0.31278926702684569301e0;
+ v[3][5][3] = 0.27141910362887187862e0;
+ v[3][5][4] = 0.24229245314748740263e0;
+ v[3][5][5] = 0.22026586649771582089e0;
+ }
+ return v[vertex_index][i][j];
+ }
--- /dev/null
+
+DEAL::
+DEAL::Calculation of the integral of f(x,y)*1/R on [0,1]x[0,1]
+DEAL::for f(x,y) = x^i y^j, with i,j ranging from 0 to 5, and R being
+DEAL::the distance from (x,y) to four vertices of the square.
+DEAL::
+DEAL:: =========Quadrature Order: 1 ===============================
+DEAL:: ============================================================
+DEAL:: ===============Vertex: 0.000000 0.000000 =============================
+DEAL::f(x,y) = x^0 y^0, Errors = 1.419233, -0.062529
+DEAL::f(x,y) = x^0 y^1, Errors = -0.250046, -0.046676
+DEAL::f(x,y) = x^0 y^2, Errors = -0.332879, -0.133607
+DEAL::f(x,y) = x^0 y^3, Errors = -0.262944, -0.155343
+DEAL::f(x,y) = x^0 y^4, Errors = -0.206246, -0.152327
+DEAL::f(x,y) = x^0 y^5, Errors = -0.167904, -0.141111
+DEAL::f(x,y) = x^1 y^0, Errors = -0.250046, -0.046676
+DEAL::f(x,y) = x^1 y^1, Errors = -0.226424, -0.100079
+DEAL::f(x,y) = x^1 y^2, Errors = -0.163944, -0.107911
+DEAL::f(x,y) = x^1 y^3, Errors = -0.121118, -0.096111
+DEAL::f(x,y) = x^1 y^4, Errors = -0.094562, -0.082873
+DEAL::f(x,y) = x^1 y^5, Errors = -0.077252, -0.071600
+DEAL::f(x,y) = x^2 y^0, Errors = -0.332879, -0.133607
+DEAL::f(x,y) = x^2 y^1, Errors = -0.163944, -0.107911
+DEAL::f(x,y) = x^2 y^2, Errors = -0.105791, -0.088336
+DEAL::f(x,y) = x^2 y^3, Errors = -0.076850, -0.070501
+DEAL::f(x,y) = x^2 y^4, Errors = -0.060010, -0.057353
+DEAL::f(x,y) = x^2 y^5, Errors = -0.049130, -0.047911
+DEAL::f(x,y) = x^3 y^0, Errors = -0.262944, -0.155343
+DEAL::f(x,y) = x^3 y^1, Errors = -0.121118, -0.096111
+DEAL::f(x,y) = x^3 y^2, Errors = -0.076850, -0.070501
+DEAL::f(x,y) = x^3 y^3, Errors = -0.055777, -0.053901
+DEAL::f(x,y) = x^3 y^4, Errors = -0.043624, -0.042958
+DEAL::f(x,y) = x^3 y^5, Errors = -0.035766, -0.035490
+DEAL::f(x,y) = x^4 y^0, Errors = -0.206246, -0.152327
+DEAL::f(x,y) = x^4 y^1, Errors = -0.094562, -0.082873
+DEAL::f(x,y) = x^4 y^2, Errors = -0.060010, -0.057353
+DEAL::f(x,y) = x^4 y^3, Errors = -0.043624, -0.042958
+DEAL::f(x,y) = x^4 y^4, Errors = -0.034164, -0.033969
+DEAL::f(x,y) = x^4 y^5, Errors = -0.028037, -0.027968
+DEAL::f(x,y) = x^5 y^0, Errors = -0.167904, -0.141111
+DEAL::f(x,y) = x^5 y^1, Errors = -0.077252, -0.071600
+DEAL::f(x,y) = x^5 y^2, Errors = -0.049130, -0.047911
+DEAL::f(x,y) = x^5 y^3, Errors = -0.035766, -0.035490
+DEAL::f(x,y) = x^5 y^4, Errors = -0.028037, -0.027968
+DEAL::f(x,y) = x^5 y^5, Errors = -0.023024, -0.023004
+DEAL:: =========Quadrature Order: 2 ===============================
+DEAL:: ============================================================
+DEAL:: ===============Vertex: 0.000000 0.000000 =============================
+DEAL::f(x,y) = x^0 y^0, Errors = 0.083005, -0.001795
+DEAL::f(x,y) = x^0 y^1, Errors = 0.034615, -0.002254
+DEAL::f(x,y) = x^0 y^2, Errors = -0.049939, -0.003643
+DEAL::f(x,y) = x^0 y^3, Errors = -0.105989, -0.004860
+DEAL::f(x,y) = x^0 y^4, Errors = -0.126977, -0.011516
+DEAL::f(x,y) = x^0 y^5, Errors = -0.128734, -0.019894
+DEAL::f(x,y) = x^1 y^0, Errors = 0.034615, -0.002254
+DEAL::f(x,y) = x^1 y^1, Errors = 0.027031, -0.002407
+DEAL::f(x,y) = x^1 y^2, Errors = -0.021731, -0.003410
+DEAL::f(x,y) = x^1 y^3, Errors = -0.049085, -0.007693
+DEAL::f(x,y) = x^1 y^4, Errors = -0.058941, -0.012832
+DEAL::f(x,y) = x^1 y^5, Errors = -0.059742, -0.017132
+DEAL::f(x,y) = x^2 y^0, Errors = -0.049939, -0.003643
+DEAL::f(x,y) = x^2 y^1, Errors = -0.021731, -0.003410
+DEAL::f(x,y) = x^2 y^2, Errors = -0.033894, -0.006861
+DEAL::f(x,y) = x^2 y^3, Errors = -0.041297, -0.010777
+DEAL::f(x,y) = x^2 y^4, Errors = -0.042534, -0.014116
+DEAL::f(x,y) = x^2 y^5, Errors = -0.040552, -0.016356
+DEAL::f(x,y) = x^3 y^0, Errors = -0.105989, -0.004860
+DEAL::f(x,y) = x^3 y^1, Errors = -0.049085, -0.007693
+DEAL::f(x,y) = x^3 y^2, Errors = -0.041297, -0.010777
+DEAL::f(x,y) = x^3 y^3, Errors = -0.038301, -0.013245
+DEAL::f(x,y) = x^3 y^4, Errors = -0.035046, -0.014991
+DEAL::f(x,y) = x^3 y^5, Errors = -0.031558, -0.015882
+DEAL::f(x,y) = x^4 y^0, Errors = -0.126977, -0.011516
+DEAL::f(x,y) = x^4 y^1, Errors = -0.058941, -0.012832
+DEAL::f(x,y) = x^4 y^2, Errors = -0.042534, -0.014116
+DEAL::f(x,y) = x^4 y^3, Errors = -0.035046, -0.014991
+DEAL::f(x,y) = x^4 y^4, Errors = -0.029956, -0.015454
+DEAL::f(x,y) = x^4 y^5, Errors = -0.025973, -0.015409
+DEAL::f(x,y) = x^5 y^0, Errors = -0.128734, -0.019894
+DEAL::f(x,y) = x^5 y^1, Errors = -0.059742, -0.017132
+DEAL::f(x,y) = x^5 y^2, Errors = -0.040552, -0.016356
+DEAL::f(x,y) = x^5 y^3, Errors = -0.031558, -0.015882
+DEAL::f(x,y) = x^5 y^4, Errors = -0.025973, -0.015409
+DEAL::f(x,y) = x^5 y^5, Errors = -0.022011, -0.014742
+DEAL:: =========Quadrature Order: 3 ===============================
+DEAL:: ============================================================
+DEAL:: ===============Vertex: 0.000000 0.000000 =============================
+DEAL::f(x,y) = x^0 y^0, Errors = 0.049500, -0.000052
+DEAL::f(x,y) = x^0 y^1, Errors = -0.000455, -0.000092
+DEAL::f(x,y) = x^0 y^2, Errors = 0.000338, -0.000200
+DEAL::f(x,y) = x^0 y^3, Errors = -0.010300, -0.000369
+DEAL::f(x,y) = x^0 y^4, Errors = -0.027273, -0.000603
+DEAL::f(x,y) = x^0 y^5, Errors = -0.042569, -0.000866
+DEAL::f(x,y) = x^1 y^0, Errors = -0.000455, -0.000092
+DEAL::f(x,y) = x^1 y^1, Errors = 0.000150, -0.000105
+DEAL::f(x,y) = x^1 y^2, Errors = 0.001770, -0.000183
+DEAL::f(x,y) = x^1 y^3, Errors = -0.004425, -0.000322
+DEAL::f(x,y) = x^1 y^4, Errors = -0.012927, -0.000525
+DEAL::f(x,y) = x^1 y^5, Errors = -0.020210, -0.000953
+DEAL::f(x,y) = x^2 y^0, Errors = 0.000338, -0.000200
+DEAL::f(x,y) = x^2 y^1, Errors = 0.001770, -0.000183
+DEAL::f(x,y) = x^2 y^2, Errors = 0.002045, -0.000229
+DEAL::f(x,y) = x^2 y^3, Errors = -0.002509, -0.000337
+DEAL::f(x,y) = x^2 y^4, Errors = -0.008202, -0.000658
+DEAL::f(x,y) = x^2 y^5, Errors = -0.012954, -0.001202
+DEAL::f(x,y) = x^3 y^0, Errors = -0.010300, -0.000369
+DEAL::f(x,y) = x^3 y^1, Errors = -0.004425, -0.000322
+DEAL::f(x,y) = x^3 y^2, Errors = -0.002509, -0.000337
+DEAL::f(x,y) = x^3 y^3, Errors = -0.004742, -0.000553
+DEAL::f(x,y) = x^3 y^4, Errors = -0.008084, -0.000986
+DEAL::f(x,y) = x^3 y^5, Errors = -0.010953, -0.001572
+DEAL::f(x,y) = x^4 y^0, Errors = -0.027273, -0.000603
+DEAL::f(x,y) = x^4 y^1, Errors = -0.012927, -0.000525
+DEAL::f(x,y) = x^4 y^2, Errors = -0.008202, -0.000658
+DEAL::f(x,y) = x^4 y^3, Errors = -0.008084, -0.000986
+DEAL::f(x,y) = x^4 y^4, Errors = -0.009419, -0.001464
+DEAL::f(x,y) = x^4 y^5, Errors = -0.010761, -0.002024
+DEAL::f(x,y) = x^5 y^0, Errors = -0.042569, -0.000866
+DEAL::f(x,y) = x^5 y^1, Errors = -0.020210, -0.000953
+DEAL::f(x,y) = x^5 y^2, Errors = -0.012954, -0.001202
+DEAL::f(x,y) = x^5 y^3, Errors = -0.010953, -0.001572
+DEAL::f(x,y) = x^5 y^4, Errors = -0.010761, -0.002024
+DEAL::f(x,y) = x^5 y^5, Errors = -0.010963, -0.002508
+DEAL:: =========Quadrature Order: 4 ===============================
+DEAL:: ============================================================
+DEAL:: ===============Vertex: 0.000000 0.000000 =============================
+DEAL::f(x,y) = x^0 y^0, Errors = 0.021072, -0.000002
+DEAL::f(x,y) = x^0 y^1, Errors = 0.000947, -0.000003
+DEAL::f(x,y) = x^0 y^2, Errors = -0.000061, -0.000009
+DEAL::f(x,y) = x^0 y^3, Errors = 0.000186, -0.000022
+DEAL::f(x,y) = x^0 y^4, Errors = -0.001562, -0.000045
+DEAL::f(x,y) = x^0 y^5, Errors = -0.005915, -0.000080
+DEAL::f(x,y) = x^1 y^0, Errors = 0.000947, -0.000003
+DEAL::f(x,y) = x^1 y^1, Errors = 0.000090, -0.000004
+DEAL::f(x,y) = x^1 y^2, Errors = -0.000191, -0.000008
+DEAL::f(x,y) = x^1 y^3, Errors = -0.000007, -0.000019
+DEAL::f(x,y) = x^1 y^4, Errors = -0.000944, -0.000038
+DEAL::f(x,y) = x^1 y^5, Errors = -0.003100, -0.000069
+DEAL::f(x,y) = x^2 y^0, Errors = -0.000061, -0.000009
+DEAL::f(x,y) = x^2 y^1, Errors = -0.000191, -0.000008
+DEAL::f(x,y) = x^2 y^2, Errors = -0.000083, -0.000011
+DEAL::f(x,y) = x^2 y^3, Errors = 0.000084, -0.000019
+DEAL::f(x,y) = x^2 y^4, Errors = -0.000575, -0.000036
+DEAL::f(x,y) = x^2 y^5, Errors = -0.002016, -0.000063
+DEAL::f(x,y) = x^3 y^0, Errors = 0.000186, -0.000022
+DEAL::f(x,y) = x^3 y^1, Errors = -0.000007, -0.000019
+DEAL::f(x,y) = x^3 y^2, Errors = 0.000084, -0.000019
+DEAL::f(x,y) = x^3 y^3, Errors = 0.000171, -0.000025
+DEAL::f(x,y) = x^3 y^4, Errors = -0.000360, -0.000039
+DEAL::f(x,y) = x^3 y^5, Errors = -0.001451, -0.000070
+DEAL::f(x,y) = x^4 y^0, Errors = -0.001562, -0.000045
+DEAL::f(x,y) = x^4 y^1, Errors = -0.000944, -0.000038
+DEAL::f(x,y) = x^4 y^2, Errors = -0.000575, -0.000036
+DEAL::f(x,y) = x^4 y^3, Errors = -0.000360, -0.000039
+DEAL::f(x,y) = x^4 y^4, Errors = -0.000688, -0.000057
+DEAL::f(x,y) = x^4 y^5, Errors = -0.001478, -0.000099
+DEAL::f(x,y) = x^5 y^0, Errors = -0.005915, -0.000080
+DEAL::f(x,y) = x^5 y^1, Errors = -0.003100, -0.000069
+DEAL::f(x,y) = x^5 y^2, Errors = -0.002016, -0.000063
+DEAL::f(x,y) = x^5 y^3, Errors = -0.001451, -0.000070
+DEAL::f(x,y) = x^5 y^4, Errors = -0.001478, -0.000099
+DEAL::f(x,y) = x^5 y^5, Errors = -0.001951, -0.000155
+DEAL:: =========Quadrature Order: 5 ===============================
+DEAL:: ============================================================
+DEAL:: ===============Vertex: 0.000000 0.000000 =============================
+DEAL::f(x,y) = x^0 y^0, Errors = 0.009674, 0.000000
+DEAL::f(x,y) = x^0 y^1, Errors = 0.000228, 0.000000
+DEAL::f(x,y) = x^0 y^2, Errors = -0.000007, 0.000000
+DEAL::f(x,y) = x^0 y^3, Errors = -0.000038, -0.000001
+DEAL::f(x,y) = x^0 y^4, Errors = -0.000015, -0.000003
+DEAL::f(x,y) = x^0 y^5, Errors = -0.000326, -0.000006
+DEAL::f(x,y) = x^1 y^0, Errors = 0.000228, 0.000000
+DEAL::f(x,y) = x^1 y^1, Errors = 0.000081, 0.000000
+DEAL::f(x,y) = x^1 y^2, Errors = 0.000029, 0.000000
+DEAL::f(x,y) = x^1 y^3, Errors = -0.000001, -0.000001
+DEAL::f(x,y) = x^1 y^4, Errors = 0.000013, -0.000002
+DEAL::f(x,y) = x^1 y^5, Errors = -0.000154, -0.000005
+DEAL::f(x,y) = x^2 y^0, Errors = -0.000007, 0.000000
+DEAL::f(x,y) = x^2 y^1, Errors = 0.000029, 0.000000
+DEAL::f(x,y) = x^2 y^2, Errors = -0.000006, 0.000000
+DEAL::f(x,y) = x^2 y^3, Errors = -0.000019, -0.000001
+DEAL::f(x,y) = x^2 y^4, Errors = -0.000005, -0.000002
+DEAL::f(x,y) = x^2 y^5, Errors = -0.000118, -0.000005
+DEAL::f(x,y) = x^3 y^0, Errors = -0.000038, -0.000001
+DEAL::f(x,y) = x^3 y^1, Errors = -0.000001, -0.000001
+DEAL::f(x,y) = x^3 y^2, Errors = -0.000019, -0.000001
+DEAL::f(x,y) = x^3 y^3, Errors = -0.000017, -0.000001
+DEAL::f(x,y) = x^3 y^4, Errors = -0.000002, -0.000002
+DEAL::f(x,y) = x^3 y^5, Errors = -0.000087, -0.000004
+DEAL::f(x,y) = x^4 y^0, Errors = -0.000015, -0.000003
+DEAL::f(x,y) = x^4 y^1, Errors = 0.000013, -0.000002
+DEAL::f(x,y) = x^4 y^2, Errors = -0.000005, -0.000002
+DEAL::f(x,y) = x^4 y^3, Errors = -0.000002, -0.000002
+DEAL::f(x,y) = x^4 y^4, Errors = 0.000009, -0.000003
+DEAL::f(x,y) = x^4 y^5, Errors = -0.000062, -0.000005
+DEAL::f(x,y) = x^5 y^0, Errors = -0.000326, -0.000006
+DEAL::f(x,y) = x^5 y^1, Errors = -0.000154, -0.000005
+DEAL::f(x,y) = x^5 y^2, Errors = -0.000118, -0.000005
+DEAL::f(x,y) = x^5 y^3, Errors = -0.000087, -0.000004
+DEAL::f(x,y) = x^5 y^4, Errors = -0.000062, -0.000005
+DEAL::f(x,y) = x^5 y^5, Errors = -0.000110, -0.000007
+DEAL:: =========Quadrature Order: 6 ===============================
+DEAL:: ============================================================
+DEAL:: ===============Vertex: 0.000000 0.000000 =============================
+DEAL::f(x,y) = x^0 y^0, Errors = 0.004879, 0.000000
+DEAL::f(x,y) = x^0 y^1, Errors = 0.000082, 0.000000
+DEAL::f(x,y) = x^0 y^2, Errors = -0.000001, 0.000000
+DEAL::f(x,y) = x^0 y^3, Errors = -0.000001, 0.000000
+DEAL::f(x,y) = x^0 y^4, Errors = -0.000004, 0.000000
+DEAL::f(x,y) = x^0 y^5, Errors = -0.000008, 0.000000
+DEAL::f(x,y) = x^1 y^0, Errors = 0.000082, 0.000000
+DEAL::f(x,y) = x^1 y^1, Errors = 0.000009, 0.000000
+DEAL::f(x,y) = x^1 y^2, Errors = 0.000001, 0.000000
+DEAL::f(x,y) = x^1 y^3, Errors = 0.000000, 0.000000
+DEAL::f(x,y) = x^1 y^4, Errors = -0.000003, 0.000000
+DEAL::f(x,y) = x^1 y^5, Errors = -0.000003, 0.000000
+DEAL::f(x,y) = x^2 y^0, Errors = -0.000001, 0.000000
+DEAL::f(x,y) = x^2 y^1, Errors = 0.000001, 0.000000
+DEAL::f(x,y) = x^2 y^2, Errors = 0.000004, 0.000000
+DEAL::f(x,y) = x^2 y^3, Errors = 0.000002, 0.000000
+DEAL::f(x,y) = x^2 y^4, Errors = 0.000000, 0.000000
+DEAL::f(x,y) = x^2 y^5, Errors = 0.000000, 0.000000
+DEAL::f(x,y) = x^3 y^0, Errors = -0.000001, 0.000000
+DEAL::f(x,y) = x^3 y^1, Errors = 0.000000, 0.000000
+DEAL::f(x,y) = x^3 y^2, Errors = 0.000002, 0.000000
+DEAL::f(x,y) = x^3 y^3, Errors = 0.000000, 0.000000
+DEAL::f(x,y) = x^3 y^4, Errors = -0.000002, 0.000000
+DEAL::f(x,y) = x^3 y^5, Errors = -0.000001, 0.000000
+DEAL::f(x,y) = x^4 y^0, Errors = -0.000004, 0.000000
+DEAL::f(x,y) = x^4 y^1, Errors = -0.000003, 0.000000
+DEAL::f(x,y) = x^4 y^2, Errors = 0.000000, 0.000000
+DEAL::f(x,y) = x^4 y^3, Errors = -0.000002, 0.000000
+DEAL::f(x,y) = x^4 y^4, Errors = -0.000002, 0.000000
+DEAL::f(x,y) = x^4 y^5, Errors = -0.000002, 0.000000
+DEAL::f(x,y) = x^5 y^0, Errors = -0.000008, 0.000000
+DEAL::f(x,y) = x^5 y^1, Errors = -0.000003, 0.000000
+DEAL::f(x,y) = x^5 y^2, Errors = 0.000000, 0.000000
+DEAL::f(x,y) = x^5 y^3, Errors = -0.000001, 0.000000
+DEAL::f(x,y) = x^5 y^4, Errors = -0.000002, 0.000000
+DEAL::f(x,y) = x^5 y^5, Errors = -0.000001, 0.000000