三维偏微分方程的快速整体谱方法
Fast global spectral methods for three-dimensional partial differential equations
摘要：
整体谱方法提供了计算偏微分方程的解的潜力，数值计算非常高的精度。在这项工作中，我们开发了一个新的整体谱方法的线性偏微分方程的立方体上的扩展的想法Chebop 2 [Townsend和Olver，J。物理、299（2015）]到利用张量化多项式基中的展开的三维设置。求解离散偏微分方程涉及到一个线性系，它可以被转换为线性张量方程。在适当的附加假设下，这些方程的结构允许通过分块递归求解器[Chen和Kressner，Numer.算法，84（2020）]。在一般情况下，当这些假设不满足时，该求解器用作预处理器以加速计算。
英文摘要：
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.
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