Cross-posting from Zulip’s #meshes.jl channel…

Does anyone know of a good reference for integration rules over different geometries?

For example, to integrate a function f over a triangle, we can consider the surface integral:

\int_S fdS = \int_{u=0}^1 \int_{v=0}^{1-u} f(u,v) ||r_u \times r_v|| dudv = Area\ \frac{f_1 + f_2 + f_3}{3}

where r = (x(u,v), y(u,v), z(u,v)) is a point in the triangle expressed in terms of parametric coordinates u and v.

More specifically, I am looking for formulas for bilinear integration over quadrangles and trilinear integration over hexahedron in terms of the vertex values f_1, f_2, f_3, ... and their coordinates x_1, x_2, ....