  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 stable decomposition
property.
