Whitney forms and their extensions

Abstract

Whitney forms are widely known as finite elements for differential forms. Whitney’s original definition yields first order functions on simplicial complexes, and a lot of research has been devoted to extending the definition to nonsimplicial cells and higher order functions. As a result, the term Whitney forms has become somewhat ambiguous in the literature. Our aim here is to clarify the concept of Whitney forms and explicitly explain their key properties. We discuss Whitney’s initial definition with more depth than usually, giving three equivalent ways to define Whitney forms. We give a comprehensive exposition of their main properties, including the proofs. Understanding of these properties is important as they can be taken as a guideline on how to extend Whitney forms to nonsimplicial cells or higher order functions. We discuss several generalisations of Whitney forms and check which of the properties can be preserved.