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The purpose of this paper is to discuss the construction of a linearoperator, referred to as the bubble transform, which maps scalar functionsdefined on a bounded domain $\Omega$ in $\mathbb{R}^n$ into a collection offunctions with local support. In fact, for a given simplicial triangulation of$\Omega$, the associated bubble transform produces a decomposition of functionson $\Omega$ into a sum of functions with support on the correspondingmacroelements. The transform is bounded in both $L^2$ and the Sobolev space$H^1$, it is local, and it preserves the corresponding continuous piecewisepolynomial spaces. As a consequence, this transform is a useful tool forconstructing local projection operators into finite element spaces such thatthe appropriate operator norms are bounded independently of polynomial degree.The transform is basically constructed by two families of operators, localaveraging operators and rational trace preserving cut--off operators.

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