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The purpose of this paper is to discuss a generalization of the bubble transform to differential forms. The bubble transform was discussed in Falk and Winther (Found Comput Math 16(1):297–328, 2016) for scalar valued functions, or zero-forms, and represents a new tool for the understanding of finite element spaces of arbitrary polynomial degree. The present paper contains a similar study for differential forms. From a simplicial mesh T of the domain Ω , we build a map which decomposes piecewise smooth k-forms into a sum of local bubbles supported on appropriate macroelements. The key properties of the decomposition are that it commutes with the exterior derivative and preserves the piecewise polynomial structure of the standard finite element spaces of k-forms. Furthermore, the transform is bounded in L 2 and also on the appropriate subspace consisting of k-forms with exterior derivatives in L 2 . [ABSTRACT FROM AUTHOR]

The purpose of this paper is to discuss the construction of a linear operator, referred to as the bubble transform, which maps scalar functions defined on into a collection of functions with local support. In fact, for a given simplicial triangulation of , the associated bubble transform produces a decomposition of functions on [Formula omitted] into a sum of functions with support on the corresponding macroelements. The transform is bounded in both and the Sobolev space , it is local, and it preserves the corresponding continuous piecewise polynomial spaces. As a consequence, this transform is a useful tool for constructing local projection operators into finite element spaces such that the appropriate operator norms are bounded independently of polynomial degree. The transform is basically constructed by two families of operators, local averaging operators and rational trace preserving cutoff operators., Author(s): Richard S. Falk[sup.1] , Ragnar Winther[sup.2] Author Affiliations: (1) Department of Mathematics, Rutgers University, 08854, Piscataway, NJ, USA (2) Department of Mathematics, University of Oslo, 0316, Oslo, Norway Introduction [...]