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