Uniform Poincar\'e inequalities for the Discrete de Rham complex on general domains

TL;DR

This paper establishes unified Poincaré inequalities for the Discrete de Rham complex on general polyhedral domains, extending previous results and providing new proofs, with applications to magnetostatics stability.

Contribution

It unifies and extends Poincaré inequalities for all operators in the DDR sequence on complex domains, including new proofs and topological generalizations.

Findings

Derived new Poincaré inequalities for gradient, divergence, and curl operators.

Extended curl inequality to domains with arbitrary second Betti numbers.

Applied inequalities to analyze stability of a DDR scheme for magnetostatics.

Abstract

In this paper we prove Poincar\'e inequalities for the Discrete de Rham (DDR) sequence on a general connected polyhedral domain $\Omega$ of $\mathbb{R}^3$. We unify the ideas behind the inequalities for all three operators in the sequence, deriving new proofs for the Poincar\'e inequalities for the gradient and the divergence, and extending the available Poincar\'e inequality for the curl to domains with arbitrary second Betti numbers. A key preliminary step consists in deriving "mimetic" Poincar\'e inequalities giving the existence and stability of the solutions to topological balance problems useful in general discrete geometric settings. As an example of application, we study the stability of a novel DDR scheme for the magnetostatics problem on domains with general topology.