为环形域谱方法构建半经典雅可比多项式的层级结构
Building hierarchies of semiclassical Jacobi polynomials for spectral methods in annuli
AI总结：
本研究构建了半经典雅可比多项式的层级结构，实现了相关矩阵的最优复杂度构造与拟最优复杂度运算，将其应用于广义Zernike环形多项式，相比Chebyshev–Fourier级数收敛更快、条件数更优，还结合圆盘与环形单元构建了高效求解径向不连续PDE的稀疏谱元法。
摘要：
我们讨论了用于计算的（可能带权的）半经典雅可比多项式族的层级结构，这类层级结构出现在多元正交多项式的构造过程中。具体而言，我们概述了如何以最优复杂度构建连接矩阵和微分矩阵，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环形域正交多项式构造中的一项特定应用——广义Zernike环形多项式，该多项式可实现偏微分方程的稀疏离散化。我们将其与缩放平移后的Chebyshev–Fourier级数进行了对比，结果表明在逼近光滑函数时，环形多项式通常收敛更快，且条件数更优。我们还结合圆盘单元和环形单元构建了一种稀疏谱元法，该方法在求解具有径向不连续变系数和数据的偏微分方程时极为高效。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
