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

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.