A Sparse Spectral Method on Triangles
A Sparse Spectral Method on Triangles
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DOI：
10.1137/19m1245888
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发表时间：
2019-01
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影响因子：
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通讯作者：
S. Olver;Alex Townsend;G. Vasil
中科院分区：
文献类型：
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作者：
S. Olver;Alex Townsend;G. Vasil
在本文中，我们证明了许多单变量正交多项式的计算工具有类似的一个家庭的二元正交多项式的三角形，包括Clenshaw的算法和稀疏微分算子。这使我们能够得到一个实用的谱方法求解线性偏微分方程的三角形稀疏离散。因此，我们可以快速解决偏微分方程使用多项式的次数在数千，导致稀疏离散多达数百万自由度。
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.