Quadrature rule over triangles

Question

I am looking for a quadrature rule of type $$ \int_T fdx\approx \sum_k f(x_k)w_k ,$$ where $T \subset \mathbb{R}^2$ is a triangle. I need the quadrature rule to be exact for polynomials of degree $\le 4$. I tried solving the weights $w_k$ and points $x_k$ by calculator, but the running time was too long. I am only looking for the rule; I don't need to know the derivation. In my problem, $T$ is the right triangle defined by vertices $(0,0)$, $(1,0)$ and $(0,1)$.

Answer 1

The link in the first comment at Numerical integration in 2D over a triangle - Quadrature formula leads to the article High degree efficient symmetrical Gaussian quadrature rules for the triangle by D.A. Dunavant, which on p. $1140$ provides the following weights and points in normalized barycentric coordinates for a quadrature rule of degree $4$ (i.e. order $5$): $$ \begin{array}{c|cc} \text{weight}&\text{points}\\\hline 0.223381589678011&(0.108103018168070,0.445948490915965,0.445948490915965)\\ 0.109951743655322&(0.816847572980459,0,091576213509771,0,091576213509771)\\ \end{array} $$ Each row represents three points on the medians, with one of the three coordinate tuples specified.

The coordinates in your coordinate system are obtained by multiplying the barycentric coordinates by the coordinates of your vertices, so you can take the first and second barycentric coordinates as your $x$ and $y$ coordinate, respectively. You need to scale the weights by $\frac12$ so they add up to the area of your triangle.

This is the next rule that would have been obtained if I’d continued the sequence I derived at Deriving quadrature on an equilateral triangle if you don't know the points. I only continued up to order $4$ there; I doubt that the weights and points for the higher orders can be obtained in closed form.