Poincaré–Friedrichs type constants for operators involving grad, curl, and div: Theory and numerical experiments

Abstract

We give some theoretical as well as computational results on Laplace and Maxwell constants, i.e., on the smallest constants cn>0 arising in estimates of the form |u|L2(Ω)≤c0|gradu|L2(Ω),|E|L2(Ω)≤c1|curlE|L2(Ω),|H|L2(Ω)≤c2|divH|L2(Ω).Besides the classical de Rham complex we investigate the complex of elasticity and the complex related to the biharmonic equation and general relativity as well using the general functional analytical concept of Hilbert complexes. We consider mixed boundary conditions and bounded Lipschitz domains of arbitrary topology. Our numerical aspects are presented by examples for the de Rham complex in 2D and 3D which not only confirm our theoretical findings but also indicate some interesting conjectures.