In one dimension there are two types of quadrature schemes.
- asymmetric rules like Newton-Cotes like formulas (Trapezodi, Simpson), and Clenshaw-Curtis place sampling points on boundary of the interval.
- 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:
- Polyquad quadpy looks great but it is all symmetric
- in this article symetric rules are also develped
- There is what I want for 2D trinagle (not for 3D tetrahedron)
- Do you know some asymetric rules for 3D objects like tetrahedron, pyramid etc. ?
- why symmetric rules are preferend ?