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Lower bounds for piecewise polynomial approximations of oscillatory functions

Jeffrey Galkowski

TL;DR

The paper delivers explicit, optimal lower bounds for the error in approximating oscillatory functions by piecewise polynomial spaces, with bounds that depend precisely on the polynomial degree $p$, meshwidth $h$, and frequency $k$. It develops a comprehensive framework from the reference element to manifolds, establishing that functions oscillating at scale $k$ cannot be approximated faster than the rate $(hk/p^2)^{p+1}$ in $L^2$ and at related rates in frequency-weighted norms, under regular mesh assumptions. These results apply directly to Helmholtz problems, including plane-wave scattering, revealing that fixed $p$ mandates $hk\to0$ to avoid pollution and requiring about $k^d$ degrees of freedom in $d$ dimensions. By bridging Fourier-analytic and geometric methods, the paper fills a gap in lower-bound theory for oscillatory PDE approximations and informs mesh design for high-frequency solvers in finite and boundary element methods.

Abstract

We prove lower bounds on the error incurred when approximating any oscillating function using piecewise polynomial spaces. The estimates are explicit in the polynomial degree and have optimal dependence on the meshwidth and frequency when the polynomial degree is fixed. These lower bounds, for example, apply when approximating solutions to Helmholtz plane wave scattering problem.

Lower bounds for piecewise polynomial approximations of oscillatory functions

TL;DR

The paper delivers explicit, optimal lower bounds for the error in approximating oscillatory functions by piecewise polynomial spaces, with bounds that depend precisely on the polynomial degree , meshwidth , and frequency . It develops a comprehensive framework from the reference element to manifolds, establishing that functions oscillating at scale cannot be approximated faster than the rate in and at related rates in frequency-weighted norms, under regular mesh assumptions. These results apply directly to Helmholtz problems, including plane-wave scattering, revealing that fixed mandates to avoid pollution and requiring about degrees of freedom in dimensions. By bridging Fourier-analytic and geometric methods, the paper fills a gap in lower-bound theory for oscillatory PDE approximations and informs mesh design for high-frequency solvers in finite and boundary element methods.

Abstract

We prove lower bounds on the error incurred when approximating any oscillating function using piecewise polynomial spaces. The estimates are explicit in the polynomial degree and have optimal dependence on the meshwidth and frequency when the polynomial degree is fixed. These lower bounds, for example, apply when approximating solutions to Helmholtz plane wave scattering problem.
Paper Structure (16 sections, 14 theorems, 81 equations)

This paper contains 16 sections, 14 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Definitions of meshes and polynomial spaces
  3. Lower bounds for approximations on $\mathbb{R}^d$
  4. Functions oscillating with a given frequency on a manifold
  5. Approximate $k$-oscillation and solutions of the Helmholtz equation
  6. $k$-oscillation of the bulk solution
  7. $k$-oscillation of the boundary traces
  8. Lower bounds on a manifold
  9. Ideas behind the proof of Theorem \ref{['t:ML2']}
  10. Comparison with Kolmogorov $n$-width bounds
  11. Estimates on the reference element
  12. Estimates at a fixed scale
  13. Uniform estimates at all scales
  14. Estimates on pairings in ${{\mathcal{S}}}_h$
  15. Estimates on the manifold
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $0<\Xi_L<\Xi_H$. Then there are $k_0>0$ and $c>0$ such that for all $p\in 0,1,\dots$, $k>k_0^{p+1}$, all $u\in L^2(\mathbb{R}^d)$ oscillating with frequency between $\Xi_Lk$ and $\Xi_Hk$, all $0<h<1$, and all piecewise polynomials, $v_h$, of degree $p$ on a regular mesh with scale $h$

Theorems & Definitions (47)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5: meshes for $M$
  • Remark 1.6
  • Definition 1.7: Broken Sobolev spaces
  • Remark 1.8
  • Definition 1.9: Scales of meshes for $M$
  • Definition 1.10: Coordinates for a scale of meshes
  • ...and 37 more