Title: The bubble transform: a new tool for analysis of finite element methods

Authors: Richard S. Falk and Ragnar Winther

Source: Foundations of Computational Mathematics, vol. 16 (2016), no. 1, 297-328

Abstract: The purpose of this paper is to discuss the construction of
a linear operator, referred to as the bubble transform, which maps
scalar functions defined on $\Omega \subset \R^n$ into a collection of
functions with local support. In fact, for a given simplicial
triangulation $\T$ of $\Omega$, the associated bubble transform
$\B_{\T}$ produces a decomposition of functions on $\Omega$ into a sum
of functions with support on the corresponding macroelements.  The
transform is bounded in both $L^2$ and the Sobolev space $H^1$, it is
local, and it preserves the corresponding continuous piecewise
polynomial spaces.  As a consequence, this transform is a useful tool
for constructing local projection operators into finite element spaces
such that the appropriate operator norms are bounded independently of
polynomial degree. The transform is basically constructed by two
families of operators, local averaging operators and rational trace
preserving cut--off operators.

Keywords: simplicial mesh, local decomposition of $H^1$, preservation of 
          piecewise polynomial spaces

URL: http://www.math.rutgers.edu/~falk/papers/bubble-I-rev.pdf