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

TL;DR

The paper develops a spline-based framework to construct orthonormal multivariate polynomials on polygonal domains via triangulations, enabling exact inner products and structured bases for $\mathbb{P}_d$ and its orthogonal complement. It introduces a practical minimization-based algorithm using Bernstein-Bézier mass matrices to obtain orthonormal bases, and extends this to iterative construction for $\mathbb{P}_{d+1}\ominus \mathbb{P}_d$, with extensive numerical examples. The authors then exploit these polynomials to derive polynomial-reduction and interpolation-based quadrature schemes on polygons, achieving one-point and higher-precision multi-point rules, and illustrating typical zero-patterns of the polynomials. The work integrates spline technology with orthogonal polynomial theory to yield efficient, high-precision quadrature on complex polygonal domains, addressing classical limitations of Gaussian-type cubature in nonstandard domains.

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.

Figures (14)

• Figure 1: Quadratic Orthonormal Polynomials over L-shaped Domain
• Figure 2: Quadratic Orthogonal Polynomials over Domain $[0, 1]^2$
• Figure 3: Contours of Quadratic Orthogonal Polynomials over Domain $[0, 1]^2$
• Figure 4: Contours of Orthogonal Polynomials of Degree 1 (top row) and Degree 2 (bottom row) over Domain $[0, 1]^2$
• Figure 5: Orthonormal Polynomials of degrees $0, 1, 2, 3$

Theorems & Definitions (58)

• Example 1.1
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Corollary 2.3
• Theorem 2.4
• proof
• Corollary 2.5
• Theorem 2.6