Fast Poisson solvers for spectral methods
Fast Poisson solvers for spectral methods
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DOI：
10.1093/imanum/drz034
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发表时间：
2017-10
影响因子：
2.1
通讯作者：
D. Fortunato;Alex Townsend
中科院分区：
文献类型：
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作者：
D. Fortunato;Alex Townsend
泊松方程是典型的椭圆型偏微分方程。虽然存在有限差分（FD）和有限元方法的快速泊松求解器，但谱方法的快速泊松求解器仍然难以捉摸。在这里，我们推导出谱方法求解泊松方程的正方形，圆柱体，实心球和立方体，具有最佳的复杂性（多对数项）的自由度来表示的解决方案。而基于FFT的快速泊松求解器利用FD矩阵的结构化特征向量，我们的求解器利用分离的光谱属性，我们精心设计的光谱离散化。如果没有并行化，我们可以在标准笔记本电脑上在2分钟内解决具有1亿个自由度的正方形上的泊松方程。
Poisson’s equation is the canonical elliptic partial differential equation. While there exist fast Poisson solvers for finite difference (FD) and finite element methods, fast Poisson solvers for spectral methods have remained elusive. Here we derive spectral methods for solving Poisson’s equation on a square, cylinder, solid sphere and cube that have optimal complexity (up to polylogarithmic terms) in terms of the degrees of freedom used to represent the solution. Whereas FFT-based fast Poisson solvers exploit structured eigenvectors of FD matrices, our solver exploits a separated spectra property that holds for our carefully designed spectral discretizations. Without parallelization we can solve Poisson’s equation on a square with 100 million degrees of freedom in under 2 min on a standard laptop.