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

Abstract

AbstractInddimensions, accurately approximating an arbitrary function oscillating with frequency$\lesssim k$≲krequires$\sim k^{d}$∼kddegrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumberk) suffers from the pollution effect if, as$k\rightarrow \infty $k→∞, the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold. While theh-version of the finite element method (FEM) (where accuracy is increased by decreasing the meshwidthhand keeping the polynomial degreepfixed) suffers from the pollution effect, thehp-FEM (where accuracy is increased by decreasing the meshwidthhand increasing the polynomial degreep) does not suffer from the pollution effect. The heart of the proof of this result is a PDE result splitting the solution of the Helmholtz equation into “high” and “low” frequency components. This result for the constant-coefficient Helmholtz equation in full space (i.e. in$\mathbb {R}^{d}$ℝd) was originally proved in Melenk and Sauter (Math. Comp79(272), 1871–1914, 2010). In this paper, we prove this result usingonlyintegration by parts and elementary properties of the Fourier transform. The proof in this paper is motivated by the recent proof in Lafontaine et al. (Comp. Math. Appl.113, 59–69, 2022) of this splitting for the variable-coefficient Helmholtz equation in full space use the more-sophisticated tools of semiclassical pseudodifferential operators.