Multiplicative construction of simplex quadrature rules is suboptimal
There is code in dune/geometry/quadraturerules/tensoproductquadrature.hh
that constructs quadrature rules for simplices by conically multiplying given 1d rules. However, this is suboptimal. Rules with higher order / less points can be constructed by using particular 1d Gauss-Jacobi rules. More information is given in the code itself. Quote:
* [snip] the 1D quadrature must be at least \f$\dim B\f$ orders higher
* than the quadrature for \f$B\f$.
*
* Question: If the polynomials are created via Duffy Transformation, do we
* really need a higher quadrature order?
*
* Answer (from OS): Not really. The official way is to use a Gauss-Jacobi
* rule with \f$ \alpha = \dim B, \beta = 0 \f$ for the 1D rule.
* That takes care of the term \f$ (1-z)^{\dim B} \f$ without needing
* additional orders. See for example A.H. Stroud, "Approximate Calculation
* of Multiple Integrals", Chapters 2.4 and 2.5 for details.
* If you want to use plain Gauss-Legendre you do need the additional orders.