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}