论文标题
一个简单的证据，证明$ hp $ -FEM不会遭受恒定效果的全空间helmholtz方程的污染效果
A simple proof that the $hp$-FEM does not suffer from the pollution effect for the constant-coefficient full-space Helmholtz equation
论文作者
论文摘要
在$ d $尺寸中，与频率$ \ lyssim k $振荡的任意函数需要$ \ sim k^d $自由度。如果作为$ k \ to \ infty $，一种解决helmholtz方程（带有波数$ k $）的数值方法会受到污染效果，维持准确性的自由度总比例比这种自然阈值快。 虽然有限元方法（fem）的$ h $ version（通过降低货币宽度$ h $并保持多项式程度$ p $固定的精度）受到污染效果的影响[Melenk，Sauter，2010]，[Melenk，Sauter，2011]，[Esterhazy，Melenk 2012]，Melenk 2012]，以及[Melenk 2012]，以及[Melenk and saut saut and saut and saut and saut and saut and sai. $ hp $ -FEM（通过减少网宽$ h $并增加多项式$ P $来提高准确性），适用于各种恒定的Helmholtz问题，不会遭受污染效应。 这些结果证明的核心是PDE结果，将Helmholtz方程的溶液分解为“高”和“低”频率分量。在这份说明性论文中，我们证明了在全空间中恒定的helmholtz方程（即，在$ \ mathbb {r}^d $中）的这种分裂，仅使用傅立叶变换的零件和基本属性进行集成；这与[Melenk，Sauter 2010]中此设置的证明相反，该设置在Bessel和Hankel函数上使用了某种涉及的边界。本文中的证明是由[Lafontaine，Spence，Wunsch 2022]的最新证明所激发的，以全面空间中可变的Helmholtz方程式进行了这种分裂；的确，[Lafontaine，Spence，Wunsch 2022]中的证明使用了更加熟悉的工具，以减少上面的基本系数以持续系数。
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 2022] of this splitting for the variable-coefficient Helmholtz equation in full space; indeed, the proof in [Lafontaine, Spence, Wunsch 2022] uses more-sophisticated tools that reduce to the elementary ones above for constant coefficients.
