三角域上的稀疏谱方法
A sparse spectral method on triangles
AI总结：
本文提出了一种基于三角域上双变量正交多项式族的稀疏谱方法，利用单变量正交多项式计算工具的类比，开发出求解三角域上线性偏微分方程的高效谱方法，实现大规模稀疏离散化。
摘要：
在本文中，我们证明了许多单变量正交多项式计算工具在三角域上的双变量正交多项式族中具有类比形式，包括Clenshaw算法和稀疏微分算子。这使得我们能够推导出一种实用的谱方法，用于求解三角域上的线性偏微分方程，利用稀疏离散化。我们因此可以快速求解偏微分方程，使用千次方的多项式，从而得到多达数百万自由度的稀疏离散化。
英文摘要：
In this paper, we demonstrate that many of the computational tools for univariate orthogonal polynomials have analogues for a family of bivariate orthogonal polynomials on the triangle, including Clenshaw's algorithm and sparse differentiation operators. This allows us to derive a practical spectral method for solving linear partial differential equations on triangles with sparse discretizations. We can thereby rapidly solve partial differential equations using polynomials with degrees in the thousands, resulting in sparse discretizations with as many as several million degrees of freedom.
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