Hallo, dear colleagues! I use the discontinues galerkin (DG) approach to obtain approximation for PDE in 3D case. So I need to integrate over the cells (tetrahedrons) and faces (triangles) with the given order of accuracy. I use the Gaussian points method for numerical integration, but it seems to me that it's not the optimal way (it's optimal only in 1D case). For example, to obtain numerical integral of the 1st degree of accuracy (that is integral is absolutely accurate for polynomials of degree k = 1) over the triangle it's enough to take only one point in the center of gravity of triangle with the weight equal to unit, while in the Gaussian points method it's necessary to take at least two points for the same accuracy. So I'm interested if there exist methods to get optimal points and weights for numerical integration in multidimensional (2D & 3D) case for different degrees of accuracy (for tetrahedrons & triangles). Tanks in advance.

You may want to read §4.6(Multidimensional Integrals) at http://www.ulib.org/webRoot/Books/Numerical_Recipes.

Unfortunately, the method described is the same as I use (gauss method or as I call it the Gaussian points method)