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：本文探讨了在多元正交多项式构建中出现的（可能带权的）半经典雅可比多项式族层次结构的计算方法。我们重点阐述了如何以最优复杂度构建连接矩阵与微分矩阵，并以拟最优复杂度实现分析与综合运算。针对这些结果在环状区域正交多项式构造中的具体应用——即广义泽尼克环形多项式——进行了研究，此类多项式可导致偏微分方程的稀疏离散化。通过与缩放平移切比雪夫-傅里叶级数进行比较发现，环形多项式在逼近光滑函数时通常具有更快的收敛速度与更好的条件数。此外，通过结合圆盘单元与环形单元构建了一种稀疏谱元方法，该方法在求解径向间断变系数与数据的偏微分方程时表现出高效性。