A Bivariate Spline Construction of Orthonormal Polynomials over Polygonal Domains and Its Applications to Quadrature

Abstract

We present computational methods for constructing orthogonal/orthonormal polynomials over arbitrary polygonal domains in $\mathbb{R}^2$ using bivariate spline functions. Leveraging a mature MATLAB implementation which generates spline spaces of any degree, any smoothness over any triangulation, we have exact polynomial representation over the polygonal domain of interest. Two algorithms are developed: one constructs orthonormal polynomials of degree $d>0$ over a polygonal domain, and the other constructs orthonormal polynomials of degree $d+1$ in the orthogonal complement of $\mathbb{P}_d$. Numerical examples for degrees $d=1--5$ illustrate the structure and zero curves of these polynomials, providing evidence against the existence of Gauss quadrature on centrally symmetric domains. In addition, we introduce polynomial reduction strategies based on odd- and even-degree orthogonal polynomials, reducing the integration to the integration of its residual quadratic or linear polynomials. These reductions motivate new quadrature schemes, which we further extend through polynomial interpolation to obtain efficient, high-precision quadrature rules for various polygonal domains.