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I am looking for highly (order 6 at least) accurate (for small triangle) quadrature formulas (using only values of the function, no derivatives) to calculate an integral of a continuous function (no singularity) over a generic triangle (or reference where I can find such formulas) :

I give you the coordinates of the three vertices (and the area of the triangle) and you give me the coordinates of the integration points and the weights associated to these points.

I found an order 3 accurate quadrature formula in the book of Quarteroni, Sacco and Saleri entitled Numerical Mathematics page 415 : if $T$ is a triangle with $a_j$ the vertices, $a^T$ the center of gravity, $a^j$ the midpoints of the edges and $|T|$ the area of $T$ then

$$ \int_T f \approx \frac{|T|}{60} \left(27f(a^T) + 3 \sum_{j=1}^3 f(a_j) + 8\sum_{j=1}^3 f(a^j) \right).$$

Do you have a similar formula for high ($\ge 6$) order of accuracy?

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Take a look at

Papanicolopulos, Stefanos-Aldo. "New fully symmetric and rotationally symmetric cubature rules on the triangle using minimal orthonormal bases." Journal of Computational and Applied Mathematics 294 (2016): 39-48.

and the papers that are cited there

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  • $\begingroup$ @Moo, There is quite a huge literature on this, and a great number of rules having various properties. There is a 6th-order method with all weights positive. To describe it involves typing in at least 200 digits. Is there a good way to do that? $\endgroup$ – Philip Roe Apr 1 '17 at 6:13
  • $\begingroup$ ArXiv preprint of the referenced paper. $\endgroup$ – ccorn Apr 1 '17 at 7:44
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I've collected a fair number of quadrature rules in quadpy, you'll find about 20 scheme families for the triangle there that match your bill.

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