A polynomial dimension-dependence analysis of Bramble--Pasciak--Xu preconditioners

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 the elliptic regularity, Bramble--Hilbert lemma, trace inequalities, and inverse inequalities. We further introduce a new quasi-interpolation operator into finite element spaces, a variant of the classical Scott--Zhang interpolation, whose 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. This result has notable implications for emerging quantum computing methodologies: recent studies indicate that polynomial dependence of BPX preconditioners on dimension can yield exponential speedups for quantum-algorithmic approaches over their classical counterparts.