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Sparse spectral and ‐finite element methods for partial differential equations on disk slices and trapeziums
Studies in Applied Mathematics ( IF 2.7 ) Pub Date : 2020-06-10 , DOI: 10.1111/sapm.12303
Ben Snowball 1 , Sheehan Olver 1
Affiliation  

Sparse spectral methods for solving partial differential equations have been derived in recent years using hierarchies of classical orthogonal polynomials on intervals, disks, and triangles. In this work we extend this methodology to a hierarchy of non-classical orthogonal polynomials on disk slices (e.g. a half-disk) and trapeziums. This builds on the observation that sparsity is guaranteed due to the boundary being defined by an algebraic curve, and that the entries of partial differential operators can be determined using formulae in terms of (non-classical) univariate orthogonal polynomials. We apply the framework to solving the Poisson, variable coefficient Helmholtz, and Biharmonic equations.

中文翻译:

磁盘切片和梯形偏微分方程的稀疏谱和有限元方法

近年来,已经使用间隔、圆盘和三角形上的经典正交多项式的层次结构推导出用于求解偏微分方程的稀疏谱方法。在这项工作中,我们将此方法扩展到磁盘切片(例如半磁盘)和梯形上的非经典正交多项式的层次结构。这建立在以下观察基础上:由于边界由代数曲线定义,因此保证了稀疏性,并且可以使用(非经典)单变量正交多项式方面的公式确定偏微分算子的条目。我们将该框架应用于求解泊松方程、可变系数亥姆霍兹方程和双调和方程。
更新日期:2020-06-10
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