On traces of the derivatives of the $L^2$-projection error

TL;DR

This work addresses bounds for derivatives at the endpoints of the $L^2$ projection error onto polynomial spaces. It develops a family of endpoint polynomials $q_{p,\nu}$ and leverages Legendre polynomials and their primitives to derive derivative trace bounds with explicit factorial factors and a growth bound $\|q_{p,\nu}\|_0^2 \le C p^{2\nu-1}$. The main result provides a general inequality $| (w-π_p w)^{(\nu)}(±1) |^2 \le \|q_{p,\nu}\|_0^2 \frac{(p-\nu-s)!}{(p+\nu+s)!} |w|_{s+\nu+1}^2$, along with a constructive recursion for $q_{p,\nu}$ and a recovered improvement over the $\nu=0$ case. These findings support rigorous error analysis for high-order finite element methods (e.g., DG, HDG, WG, HHO) and provide novel $L^2$ norms for integrated Legendre polynomials, enriching the toolbox for spectral and high-order discretizations.

Abstract

We provide derivative estimates for the $L^2$ projection of an $H^{k}$ function onto the space of polynomials of degree $\leq p$. The bounds are explicit in the order of differentiation and the polynomial degree $p$.

Theorems & Definitions (14)

• Proposition 1
• Lemma 1: $L^2$-norms of the (integrated) Legendre polynomials
• Lemma 2
• proof
• Remark 1
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 1