In $d$ dimensions, approximating an arbitrary function oscillating with frequency $\lesssim k$ requires $\sim k^d$ degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber $k$) suffers from the pollution effect if, as $k\to \infty$, 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 [Melenk, Sauter 2010], [Melenk, Sauter 2011], [Esterhazy, Melenk 2012], and [Melenk, Parsania, Sauter 2013] 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 $\mathbb{R}^d$) using only integration by parts and elementary properties of the Fourier transform; this is in contrast to the proof for this set-up in [Melenk, Sauter 2010] which uses somewhat-involved bounds on Bessel and Hankel functions. The proof in this paper is motivated by the recent proof in [Lafontaine, Spence, Wunsch 2020] of this splitting for the variable-coefficient Helmholtz equation in full space; indeed, the proof in [Lafontaine, Spence, Wunsch 2020] uses more-sophisticated tools that reduce to the elementary ones above for constant coefficients.
翻译:在 $dd美元 维基元值的维基元值上, 一个任意的函数变换 以频率变速 $\ lsmusim k$ 需要 $smlxm 自由度 。 解决 Helmholtz 等式( 以波数 $k美元 ) 的数字方法受到污染效应的影响, 如果以美元计, 维持精确度所需的总自由度比这个自然阈值增长更快 。 有限元素法( FEM) 的变换 $hzm 的精确度( 通过降低 meshwth $hmlxl, 并保持多位平流量 $pal $pal) 的精确度 。 所庆祝的论文[ Melenholt, sater, [Merlenk, 2012], [Esterhacyhazy, sater] 保持精确度所需的自由度的总量比这个自然阈值更快 。 在平基元值法中, 通过降低 平基元值的平基值 的精确度, 和平基底的平基值变化结果 。