Lower bounds for piecewise polynomial approximations of oscillatory functions

TL;DR

This paper establishes fundamental lower bounds on the approximation error for oscillatory functions using piecewise polynomials, highlighting the limitations of such methods especially in high-frequency problems like Helmholtz scattering.

Contribution

It provides explicit lower bounds that depend on polynomial degree, meshwidth, and frequency, advancing understanding of approximation limits for oscillatory functions.

Findings

Lower bounds depend explicitly on polynomial degree and meshwidth.

Bounds are optimal with respect to frequency for fixed polynomial degree.

Applicable to Helmholtz scattering solutions and similar oscillatory problems.

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.