We discuss computing with hierarchies of families of (potentially weighted) semiclassical Jacobi polynomials which arise in the construction of multivariate orthogonal polynomials. In particular, we outline how to build connection and differentiation matrices with optimal complexity and compute analysis and synthesis operations in quasi-optimal complexity. We investigate a particular application of these results to constructing orthogonal polynomials in annuli, called the generalised Zernike annular polynomials, which lead to sparse discretisations of partial differential equations. We compare against a scaled-and-shifted Chebyshev--Fourier series showing that in general the annular polynomials converge faster when approximating smooth functions and have better conditioning. We also construct a sparse spectral element method by combining disk and annulus cells, which is highly effective for solving PDEs with radially discontinuous variable coefficients and data.

翻译：本文探讨了在构建多元正交多项式过程中出现的（可能加权的）半经典雅可比多项式族层次结构的计算方法。特别地，我们概述了如何以最优复杂度构建连接矩阵与微分矩阵，并以拟最优复杂度完成分析与合成运算。我们研究了这些结果在构建环形区域正交多项式（称为广义Zernike环形多项式）中的一个具体应用，该多项式可导致偏微分方程的稀疏离散化。通过与缩放平移后的切比雪夫-傅里叶级数进行对比，我们发现环形多项式在逼近光滑函数时通常具有更快的收敛速度和更优的条件数。我们还通过组合圆盘与环形单元构建了稀疏谱元方法，该方法对于求解具有径向不连续变系数及数据的偏微分方程极为有效。