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Attached to Project - Dune
Summary - Error in simplex quadrature rules of order 6 and 7
Task Type - Bug Report
Category - Quadrature
Status - Unconfirmed
Assigned To - 
Operating System - All
Severity - Medium
Priority - Normal
Reported Version - SVN
Due in Version - 
Due Date - Undecided
Details - This is NOT a duplicate of FS#162!
The triangle quadrature rules of order 6 and 7 produce wrong results.
This is easily
checked by integrating y^p:

#include <iostream>

#include "dune/config.h"
#include "dune/grid/common/quadraturerules.hh"

int main() 
{
  using namespace Dune;
  
  for (int p=0; p<12; ++p) {
    QuadratureRule<double,2> const& qr =
QuadratureRules<double,2>::rule(GeometryType(GeometryType::simplex,2),p);
    double integ = 0;
    for (size_t g=0; g<qr.size(); ++g) {
      FieldVector<double,2> const& x = qr[g].position();
      double weight = qr[g].weight();

      // integrating x[0]^p instead yields correct result!
      // Probably the y-coordinate of an integration point is wrong.
      integ += weight*std::pow(x[1],p);
    }
    double exact = 1.0/((p+2)*(p+1));
    std::cout.precision(16);
    std::cout << "p=" << p << " integ=" << integ << " (should be " <<
exact << ") error = " << integ-exact << "n";
  }
  return 0;
}

This program produces the following output:

p=0 integ=0.5 (should be 0.5) error = 0
p=1 integ=0.16666666665 (should be 0.1666666666666667) error =
-1.666666804567285e-11
p=2 integ=0.08333333333333331 (should be 0.08333333333333333) error =
-1.387778780781446e-17
p=3 integ=0.05 (should be 0.05) error = 0
p=4 integ=0.03333333333333333 (should be 0.03333333333333333) error =
0
p=5 integ=0.02380952380952381 (should be 0.02380952380952381) error =
3.469446951953614e-18
p=6 integ=0.01788018253696759 (should be 0.01785714285714286) error =
2.303967982473024e-05
p=7 integ=0.01389590970262738 (should be 0.01388888888888889) error =
7.020813738494933e-06
p=8 integ=0.01111111111111111 (should be 0.01111111111111111) error =
1.734723475976807e-18
p=9 integ=0.009090909090909089 (should be 0.00909090909090909) error =
-1.734723475976807e-18
p=10 integ=0.00757575757575758 (should be 0.007575757575757576) error
= 4.336808689942018e-18
p=11 integ=0.006410256410256407 (should be 0.00641025641025641) error
= -2.602085213965211e-18

In particular, order 6 and 7 are quite wrong, but also order 1 seems
to be less accurate than
one should like.

Martin Weiser


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