# 3D Quadrature schemes with points on boundary

In one dimension there are two types of quadrature schemes.

1. asymmetric rules like Newton-Cotes like formulas (Trapezodi, Simpson), and Clenshaw-Curtis place sampling points on boundary of the interval.
2. symmetric Gaussian Quadrature use only internal points which never tough the boundary.

Clear advantage of asymmeric formulas with boundary points is that one can re-use the function values on the boundary of domain for neighboring domains. For example If I have tetraheral mesh I can use the values for sampling points placed on vertexes, edges and faces for all neighboring tetrahedra.

However, when I was searching in literature I mostly found symmetric 2D and 3D quadrature schemes with just internal points (especially for tetrahedra, pyramids etc.). For example:

Qustions:

• Do you know some asymetric rules for 3D objects like tetrahedron, pyramid etc. ?
• why symmetric rules are preferend ?
• Gauss-Lobatto rules include the extreme of the interval. Commented Dec 3, 2023 at 18:04
• You can't reuse if the function you are integrating has discontinuity over element boundary. In fact, you'd need to somehow distinguish between these two cases which sounds cumbersome.
– knl
Commented Dec 3, 2023 at 19:44
• @nicoguaro OK, but how are Gauss-Lobatto rules defined for tetrahedron ? Commented Dec 5, 2023 at 19:55
• @knl OK, but I don't have discontinuous functions Commented Dec 5, 2023 at 19:56