Uniform Poincar\'{e} inequalities for the discrete de Rham complex of differential forms

TL;DR

This paper establishes uniform discrete Poincaré inequalities for the discrete de Rham complex of differential forms on polytopal domains, extending known inequalities to arbitrary dimensions and topologies, with implications for numerical methods.

Contribution

It unifies and extends Poincaré inequalities for all differential operators in the discrete de Rham complex to general polytopal domains of any dimension and topology.

Findings

Proved mesh-independent Poincaré inequalities for the entire discrete de Rham complex.

Extended inequalities for gradient, curl, and divergence to arbitrary dimensions.

Provided inequalities useful for stability and existence proofs in numerical schemes.

Abstract

In this paper we prove discrete Poincar\'e inequalities that are uniform in the mesh size for the discrete de Rham complex of differential forms developed in [Bonaldi, Di Pietro, Droniou, and Hu, An exterior calculus framework for polytopal methods, J. Eur. Math. Soc., to appear, arXiv preprint 2303.11093]. We unify the underlying ideas behind the Poincar\'e inequalities for all differential operators in the sequence, extending the known inequalities for the gradient, curl, and divergence in three-dimensions to polytopal domains of arbitrary dimension and general topology. A key step in the proof involves deriving specific Poincar\'e inequalities for the cochain complex supported on the polytopal mesh. These inequalities are of independent interest, as they are useful, for instance, in establishing the existence and stability, on domains of generic topology, of solutions of schemes based on Mimetic Finite Differences, Compatible Discrete Operators or Discrete Geometric Approach.