Title:         Local space-preserving decompositions for the bubble
               transform

Authors:       Richard S. Falk and Ragnar Winther

Source:        preprint

Abstract:     

      The bubble transform is a procedure to decompose
      differential forms, which are piecewise smooth with respect to a
      given triangulation of the domain, into a sum of local bubbles.
      In this paper, an improved version of a construction in the
      setting of the de Rham complex previously proposed by the
      authors is presented.  The major improvement in the
      decomposition is that unlike the previous results, in which the
      individual bubbles were rational functions with the property
      that groups of local bubbles summed up to preserve piecewise
      smoothness, the new decomposition is strictly space-preserving
      in the sense that each local bubble preserves piecewise
      smoothness.  An important property of the transform is that the
      construction only depends on the given triangulation of the
      domain and is independent of any finite element space. On the
      other hand, all the standard piecewise polynomial spaces are
      invariant under the transform.  Other key properties of the
      transform are that it commutes with the exterior derivative, is
      bounded in $L^2$, and satisfies the {\it stable decomposition
      property}

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

AMS(MOS) subject classifications: 5N30, 52-08}