A simple proof that the hp-FEM does not suffer from the pollution effect for the constant-coefficient full-space Helmholtz equation

Abstract In d dimensions, accurately approximating an arbitrary function oscillating with frequency $\lesssim k$ ≲ k requires $\sim k^{d}$ ∼ d degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber k ) suffers from pollution effect if, as $k\rightarrow \infty $ → ∞ , total number freedom needed to maintain accuracy grows faster than this natural threshold. While h -version finite element (FEM) (where is increased by decreasing meshwidth and keeping polynomial degree p fixed) effect, hp -FEM increasing does not suffer effect. The heart proof result a PDE splitting solution into “high” “low” components. This constant-coefficient in full space (i.e. $\mathbb {R}^{d}$ ℝ was originally proved Melenk Sauter ( Math. Comp 79 (272), 1871–1914, 2010). paper, we prove using only integration parts elementary properties Fourier transform. paper motivated recent Lafontaine et al. Comp. Appl. 113 59–69, 2022) variable-coefficient use more-sophisticated tools semiclassical pseudodifferential operators.