On the stability of the boundary trace of the polynomial L^2-projection on triangles and tetrahedra (extended version)

TL;DR

This paper investigates the stability of the boundary trace of polynomial L^2-projections on triangles and tetrahedra, establishing bounds that lead to optimal convergence rates for approximation errors on these domains.

Contribution

It provides new stability estimates for the boundary trace of polynomial L^2-projections on triangles and tetrahedra, enabling improved error analysis in finite element methods.

Findings

Proves a stability inequality for the boundary trace of the L^2-projection.

Establishes optimal convergence rates for boundary approximation errors.

Provides theoretical foundations for finite element error analysis.

Abstract

For the reference triangle or tetrahedron $T$, we study the stability properties of the $L^2(T)$-projection $\Pi_N$ onto the space of polynomials of degree $N$. We show $\|\Pi_N u\|_{L^2(\partial T)}^2 \leq C \|u\|_{L^2(T)} \|u\|_{H^1(T)}$. This implies optimal convergence rates for the approximation error $\|u - \Pi_N u\|_{L^2(\partial T)}$ for all $u \in H^k(T)$, $k > 1/2$.