Dual Perturbation Method for Spectral Solution of Block‐Posed Harmonic Problems

• Published in: Numerical linear algebra with applications Vol. 32; no. 1
• Main Author: Lau, Stephen R.
• Format: Journal Article
• Language: English
• Published: Oxford Wiley Subscription Services, Inc 01.02.2025
• Subjects: Boundary conditions; Chebyshev approximation; Perturbation methods
• ISSN: 1070-5325; 1099-1506
• DOI: 10.1002/nla.2587

We present a new direct (or quasi‐direct) strategy for solving the three‐dimensional Poisson and Helmholtz problems posed on a Cartesian block subject to Dirichlet boundary conditions. Our approach starts with a spectral approximation of either problem involving modal Chebyshev integration matrices. With the number of Chebyshev modes associated with each of the coordinate directions, the total number of modes is . The relevant complexities for our base methods are then similar to certain classical methods; in particular, a set‐up cost scaling like and, thereafter, an solve cost. The memory storage for our approach is and involves no hierarchical data formats. Our approaches exhibit spectral accuracy and are empirically well‐conditioned. We describe acceleration via the introduction of an iterative element. This acceleration yields a method with an set‐up cost, followed by a sub‐quadratic solve complexity seen empirically to also be . The concluding section remarks on possible further acceleration (not established), targeting set‐up and solve costs.