2025/12/05 by Jiang, Boou, Park, Jongho, Xu, Jinchao · 1 citation
Computer Science · Engineering · Physics and Astronomy · #65F08 #65N30 #65N55 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Matrix Theory and Algorithms #Model Reduction and Neural Networks #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2512.06166
openalex publication_date 2025/12/05 · openalex created_date 2025/12/10 · openalex updated_date 2026/07/28
We investigate the dimension dependence of Bramble--Pasciak--Xu (BPX) preconditioners for high-dimensional partial differential equations and establish that the condition numbers of BPX-preconditioned systems grow only polynomially with the spatial dimension. Our analysis requires a careful derivation of the dimension dependence of several fundamental tools in the theory of finite element methods, including elliptic regularity, the Bramble--Hilbert lemma, trace inequalities, and inverse inequalities. We further analyze an averaged Scott--Zhang-type quasi-interpolation operator, and show that its associated constants scale polynomially with the dimension. Building on these ingredients, we prove a multilevel norm equivalence theorem and derive a BPX preconditioner with explicit polynomial bounds on its dimensional dependence. The analysis is motivated in part by recent tensor and quantum finite element methods, where dimension-explicit conditioning estimates for BPX preconditioners play an important role.