Poincare-Friedrichs Type Constants for Operators Involving grad, curl, and div: Theory and Numerical Experiments

TL;DR

This paper investigates Poincare-Friedrichs type constants for operators involving grad, curl, and div, providing theoretical insights and numerical experiments across various complex settings including elasticity and general relativity.

Contribution

It extends the analysis of Laplace and Maxwell constants to new complexes like elasticity and biharmonic, using Hilbert complexes and mixed boundary conditions.

Findings

Numerical experiments confirm theoretical bounds.

Examples suggest new conjectures in the field.

Analysis applies to complex geometries and topologies.

Abstract

We give some theoretical as well as computational results on Laplace and Maxwell constants. 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.