The Bubble Transform and the de Rham Complex

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 [Formula omitted] of the domain [Formula omitted], 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 [Formula omitted] and also on the appropriate subspace consisting of k-forms with exterior derivatives in [Formula omitted].
Author(s): Richard S. Falk [sup.1] , Ragnar Winther [sup.2] Author Affiliations: (1) https://ror.org/05vt9qd57, grid.430387.b, 0000 0004 1936 8796, Department of Mathematics, Rutgers University, , 08854, Piscataway, NJ, USA (2) https://ror.org/01xtthb56, [...]