Discrete Poincare and Bogovskii operators on cochains and Whitney forms

TL;DR

This paper develops explicit discrete Poincare and Bogovskii operators for cochains and Whitney forms, enabling advanced finite element exterior calculus applications on various complex domains.

Contribution

It introduces new constructive methods for discrete Poincare and Bogovskii operators applicable to simplicial complexes and Lipschitz domains.

Findings

Constructed explicit discrete Poincare operators for collapsible and star-shaped complexes.

Extended these operators to Lipschitz contractible domains.

Modified operators to satisfy homotopy identities and boundary conditions.

Abstract

Smooth Poincare operators are a tool used to show the vanishing of smooth de Rham cohomology on contractible manifolds and have found use in the analysis of finite element methods based on the Finite Element Exterior Calculus (FEEC). We construct analogous discrete Poincare operators acting on cochains and Whitney forms. We provide explicit, constructive realizations of these operators under various assumptions on the underlying domain or simplicial complex. In particular, we provide simple constructions for the discrete Poincare operators on simplicial complexes which are collapsible and those with underlying domain being star-shaped with respect to a point. We then provide more abstract constructions on simplicial complexes which are discrete contractible and domains which are Lipschitz contractible. We also modify the discrete Poincare operator on star-shaped domains to construct a discrete Bogovskii operator which satisfies the requisite homotopy identity while preserving homogeneous boundary conditions. Applications arise in the construction of discrete scalar and vector potentials and in the discrete wedge product of Discrete Exterior Calculus (DEC).