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, approximating an arbitrary function oscillating with frequency k requires ∼ k d degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber k) suffers from the pollution effect if, as k → ∞, the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold.While the h-version of the finite element method (FEM) (where accuracy is increased by decreasing the meshwidth h and keeping the polynomial degree p fixed) suffers from the pollution effect, the celebrated papers [41], [42], [17], and [40] showed that the hp-FEM (where accuracy is increased by decreasing the meshwidth h and increasing the polynomial degree p) applied to a variety of constant-coefficient Helmholtz problems does not suffer from the pollution effect.The heart of the proofs of these results is a PDE result splitting the solution of the Helmholtz equation into "high" and "low" frequency components. In this expository paper we prove this splitting for the constant-coefficient Helmholtz equation in full space (i.e., in R d ) using only integration by parts and elementary properties of the Fourier transform; this is in contrast to the proof for this set-up in [41] which uses somewhat-involved bounds on Bessel and Hankel functions. The proof in this paper is motivated by the recent proof in [34] of this splitting for the variable-coefficient Helmholtz equation in full space; indeed, the proof in [34] uses more-sophisticated tools that reduce to the elementary ones above for constant coefficients.We combine this splitting with (i) standard arguments about convergence of the FEM applied to the Helmholtz equation (the so-called "Schatz argument", which we reproduce here) and (ii) polynomial-approximation results (which we quote from the literature without proof) to give a simple proof that the hp-FEM does not suffer from the pollution effect for the constant-coefficient full-space Helmholtz equation.

Asymptotic Dispersion Correction in General Finite Difference Schemes for Helmholtz Problems

“…For the hp-Finite Element method for Helmholtz problems, it is known that the pollution effect can actually be suppressed (see e.g. [17,19,20,21,24,32]) if kh/p is small enough, and p ≥ C log(k) for a large enough constant C. Such results have been obtained for Discontinuous-Galerkin methods as well in [22].…”

“…In addition, at the continuous level, plane waves are given by e ikx•θ for θ ∈ S d−1 whereas, at the discrete level, plane waves are given by e ik d x•θ where k d is the discrete wavenumber, which depends on θ and the meshsize h, and we usually have k d (θ, h) ̸ = k, which is called the dispersion error. The dispersion error is also responsible for the pollution effect [17,20,24,32], which is the fact that keeping kh small is not enough to prevent the relative error to grow with the wavenumber.…”