Gauss–Green cubature and moment computation over arbitrary geometries

We have implemented in Matlab a Gauss-like cubature formula over arbitrary bivariate domains with a piecewise regular boundary, which is tracked by splines of maximum degree p (spline curvilinear polygons). The formula is exact for polynomials of degree at most 2 n − 1 using N ∼ c m n 2 nodes, 1 ≤ c ≤ p , m being the total number of points given on the boundary. It does not need any decomposition of the domain, but relies directly on univariate Gauss–Legendre quadrature via Green’s integral formula. Several numerical tests are presented, including computation of standard as well as orthogonal moments over a nonstandard planar region.