A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions

We present a numerical algorithm for the construction of efficient, high-order quadratures in two and higher dimensions. Quadrature rules constructed via this algorithm possess positive weights and interior nodes, resembling the Gaussian quadratures in one dimension. In addition, rules can be generated with varying degrees of symmetry, adaptable to individual domains. We illustrate the performance of our method with numerical examples, and report quadrature rules for polynomials on triangles, squares, and cubes, up to degree 50. These formulae are near optimal in the number of nodes used, and many of them appear to be new.