Discrete Poincaré and Bogovski\uı operators on cochains and Whitney forms
This work provides foundational tools for finite element methods and discrete exterior calculus, enabling discrete analogs of continuous operators used in cohomology and potential theory.
The authors construct discrete Poincaré and Bogovskiĭ operators on cochains and Whitney forms, providing explicit realizations for collapsible and star-shaped domains, and more abstract constructions for contractible simplicial complexes and Lipschitz contractible domains. These operators enable discrete scalar and vector potentials and a discrete wedge product in Discrete Exterior Calculus.
Smooth Poincaré 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 analagous discrete Poincaré 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 Poincaré 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 Poincaré operator on star-shaped domains to construct a discrete Bogovski\uı 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).