Numerical integration of homogeneous functions on convex and nonconvex polygons and polyhedra

We present a method for the numerical integration of homogeneous functions over convex and nonconvex polygons and polyhedra. On applying Stokes's theorem and using the property of homogeneous functions, we show that it suffices to integrate these functions on the boundary facets of the polytope. For homogeneous polynomials, this approach is used to further reduce the integration to just function evaluations at the vertices of the polytope. This results in an exact cubature rule for a homogeneous polynomial f, where the integration points are only the vertices of the polytope and the function f and its partial derivatives are evaluated at these vertices. Numerical integration of homogeneous functions in polar coordinates and on curved domains are also presented. Along with an efficient algorithm for its implementation, we showcase several illustrative examples in two and three dimensions that demonstrate the accuracy of the proposed method.