Computable Poincar\'e--Friedrichs constants for the $L^{p}$ de~Rham complex over convex domains and domains with shellable triangulations

TL;DR

This paper develops explicit bounds for Poincaré--Friedrichs constants and eigenvalues of vector Laplacians on convex and shellable triangulated domains, using potentials for differential operators in the de Rham complex.

Contribution

It introduces computable bounds for potentials of the exterior derivative on shellable triangulations and convex domains, enabling explicit Poincaré--Friedrichs constants and eigenvalue estimates.

Findings

Explicit bounds for operator norms of potentials depend only on geometry.

Computed Poincaré--Friedrichs constants for the $L^{p}$ de Rham complex.

Validated theoretical bounds with computational examples.

Abstract

We construct potentials for the exterior derivative, in particular, for the gradient, the curl, and the divergence operators, over domains with shellable triangulations. Notably, the class of shellable triangulations includes local patches (stars) in two or three dimensions. The operator norms of our potentials satisfy explicitly computable bounds that depend only on the geometry. We thus compute upper bounds for constants in Poincar\'e--Friedrichs inequalities and lower bounds for the eigenvalues of vector Laplacians. As an additional result with independent standing, we establish Poincar\'e--Friedrichs inequalities with computable constants for the $L^{p}$ de~Rham complex over bounded convex domains, derived as explicit operator norms of regularized Poincar\'e and Bogovski\u{\i} potential operators. We express all our main results in the calculus of differential forms and treat the gradient, curl, and divergence operators as instances of the exterior derivative. Computational examples illustrate the theoretical findings.