Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons

We construct efficient quadratures for the integration of polynomials over irregular convex polygons and polyhedrons based on moment fitting equations. The quadrature construction scheme involves the integration of monomial basis functions, which is performed using homogeneous quadratures with minimal number of integration points, and the solution of a small linear system of equations. The construction of homogeneous quadratures is based on Lasserre's method for the integration of homogeneous functions over convex polytopes. We also construct quadratures for the integration of discontinuous functions without the need to partition the domain into triangles or tetrahedrons. Several examples in two and three dimensions are presented that demonstrate the accuracy and versatility of the proposed method.