Abstract

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.

中文翻译：

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具有弱奇异核的非局部问题的分段多项式搭配的尖锐误差估计

摘要

泊松方程是规范的椭圆偏微分方程。尽管存在用于有限差分（FD）和有限元方法的快速泊松求解器，但是用于频谱方法的快速泊松求解器仍然难以捉摸。在这里，我们得出用于在正方形，圆柱体，实心球和立方体上求解泊松方程的频谱方法，这些方法在表示解的自由度方面具有最佳的复杂度（最多为多对数项）。基于FFT的快速Poisson求解器利用FD矩阵的结构化特征向量，而我们的求解器则利用了分离的光谱属性，该属性适用于我们精心设计的光谱离散化。如果没有并行化，我们可以在2分钟内用标准笔记本电脑在1亿自由度的正方形上求解泊松方程。