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 ideas of Chebop2 [Townsend and Olver, J. Comput. Phys., 299 (2015)] to the three-dimensional setting utilizing expansions in tensorized polynomial bases. Solving the discretized PDE involves a linear system that can be recast as a linear tensor equation. Under suitable additional assumptions, the structure of these equations admits for an efficient solution via the blocked recursive solver [Chen and Kressner, Numer. Algorithms, 84 (2020)]. In the general case, when these assumptions are not satisfied, this solver is used as a preconditioner to speed up computations.

翻译：全球光谱方法提供了从数字上以非常高的精确度计算部分差异方程的解决方案的可能性。 在这项工作中,我们开发了一种新型全球光谱方法,用于立方体上的线性部分差异方程,方法是将Chebop2[Townsend和Olver,J.Comput.J.Phys.,299(2015年)]的理念扩大到三维设置,利用有压力的多元基数的扩张。解析离式PDE包含一个线性系统,可以重新作为线性抗拉方程。在适当的额外假设下,这些方程的结构通过被封的递归解器[Chen和Kressner,Numer.Algorithms,84(2020年)]认可了高效的解决方案。在一般情况下,如果这些假设不能得到满足,则该求解器被用作加速计算的先决条件。