Constants in Discrete Poincaré and Friedrichs Inequalities and Discrete Quasi-Interpolation

Abstract

This paper provides a discrete Poincaré 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 $C(n)\mathrm{diam}(K)$. 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.