Fast global spectral methods for three-dimensional partial differential equations

TL;DR

This paper introduces a new global spectral method for solving three-dimensional linear partial differential equations on cubes, leveraging tensorized polynomial bases and efficient tensor solvers for high-accuracy numerical solutions.

Contribution

The paper extends spectral methods to 3D PDEs using tensorized polynomial bases and develops an efficient solver framework, including preconditioning techniques for general cases.

Findings

Efficient solution of 3D PDEs using tensor-based spectral methods.

Implementation of a recursive solver for structured tensor equations.

Preconditioning improves computational speed when assumptions are not met.

Abstract

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.