Title:         The Bubble Transform and the de Rham Complex

Authors:       Richard S. Falk and Ragnar Winther

Source:        to appear in Foundations of Computational Mathematics

Abstract:     
The purpose of this paper is to discuss a generalization of the
  bubble transform to differential forms. The bubble transform was
  discussed in [19] 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 $\Omega$, 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 $L^2$.


Key words: simplicial mesh, commuting decompostion of $k$-forms, 
preservation of piecewise polynomial spaces    

AMS(MOS) subject classifications. Primary: 65N30, 52-08