Constants in Discrete Poincar\'e and Friedrichs Inequalities and Discrete Quasi-Interpolation

TL;DR

This paper establishes explicit constants for discrete Poincaré and Friedrichs inequalities on simplices, and analyzes the stability of a quasi-interpolator for adaptive finite element methods, ensuring reliable error estimates.

Contribution

It provides explicit constants in discrete inequalities and proves the stability of a quasi-interpolator, enhancing adaptive finite element analysis.

Findings

Explicit constants for discrete Poincaré inequality depending on dimension.

Stability of an enrichment operator leading to reliable quasi-interpolation.

Bounds on constants in adaptivity axioms for optimal convergence.

Abstract

This paper provides a discrete Poincar\'e inequality in $n$ space dimensions on a simplex $K$ with explicit constants. This inequality bounds the norm of the piecewise derivative of functions with integral mean zero on $K$ and all integrals of jumps zero along all interior sides by its Lebesgue norm by $C(n)\operatorname{diam}(K)$. The explicit constant $C(n)$ depends only on the dimension $n=2,3$ in case of an adaptive triangulation with the newest vertex bisection. The second part of this paper proves the stability of an enrichment operator, which leads to the stability and approximation of a (discrete) quasi-interpolator applied in the proofs of the discrete Friedrichs inequality and discrete reliability estimate with explicit bounds on the constants in terms of the minimal angle $\omega_0$ in the triangulation. The analysis allows the bound of two constants $\Lambda_1$ and $\Lambda_3$ in the axioms of adaptivity for the practical choice of the bulk parameter with guaranteed optimal convergence rates.