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

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.