triangle_witherden_rule

triangle_witherden_rule, a Python code which returns a symmetric Witherden quadrature rule for the triangle, with exactness up to total degree 20.

The data is given for the following triangle:

      (0,1) 
        | \ 
        |  \ 
        |   \ 
        |    \ 
      (0,0)--(1,0)
      

We suppose we are given a triangle T with vertices A, B, C. We call a rule with n points, returning barycentric coordinates a, b, c, and weights w. Then the integral I of f(x,y) over T is approximated by Q as follows:

      (x,y) = a(1:n) * A + b(1:n) * B + c(1:n) * C
      Q = area(T) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i)) 
     

Licensing:

The computer code and data files made available on this web page are distributed under the MIT license

Languages:

triangle_witherden_rule is available in a C version and a C++ version and a MATLAB version and an Octave versionand a Python version.

Related Data and Programs:

triangle_integrals, a Python code which returns the exact value of the integral of any monomial over the interior of the unit triangle in 2d.

triangle_monte_carlo, a Python code which uses the Monte Carlo method to estimate the integral of a function over the interior of the unit triangle in 2d.

triangle_twb_rule, a Python code which generates the points and weights of quadrature rules over the interior of a triangle in 2D, determined by Taylor, Wingate, and Bos.

Reference:

1. Freddie Witherden, Peter Vincent, On the identification of symmetric quadrature rules for finite element methods, Computers and Mathematics with Applications, Volume 69, pages 1232-1241, 2015.

Source Code:

• triangle_witherden_rule.py, the source code.
• triangle_witherden_rule.sh, runs all the tests.
• triangle_witherden_rule.txt, the output file.