Global spectral methods offer the potential to compute solutions of partial differential equations numerically to very high accuracy. In this work, we develop a novel global spectral method for linear partial differential equations on cubes by extending the ideas of Chebop2 (Townsend, A. & Olver, S. (2015) The automatic solution of partial differential equations using a global spectral method. J. Comput. Phys., 299, 106–123) to the three-dimensional setting utilizing expansions in tensorized polynomial bases. Solving the discretized partial differential equation involves a linear system that can be recast as a linear tensor equation. Under suitable additional assumptions, the structure of these equations admits an efficient solution via the blocked recursive solver (Chen, M. & Kressner, D. (2020) Recursive blocked algorithms for linear systems with Kronecker product structure. Numer. Algorithms, 84, 1199–1216). In the general case, when these assumptions are not satisfied, this solver is used as a preconditioner to speed up computations.

中文翻译：

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三维偏微分方程的快速全局谱方法

全局谱方法提供了以非常高的精度数值计算偏微分方程解的潜力。在这项工作中，我们通过扩展 Chebop2 (Townsend, A. & Olver, S. (2015) 使用全局谱方法的偏微分方程的自动求解的思想，为立方体上的线性偏微分方程开发了一种新的全局谱方法。 J. Comput. Phys., 299, 106-123) 利用张量多项式基中的展开来进行三维设置。求解离散偏微分方程涉及一个线性系统，该系统可以重铸为线性张量方程。在适当的附加假设下，这些方程的结构允许通过阻塞递归求解器 (Chen, M. & Kressner, D. (2020) 具有 Kronecker 乘积结构的线性系统的递归阻塞算法。编号。算法, 84, 1199–1216)。在一般情况下，当这些假设不满足时，该求解器用作加速计算的预条件器。