The Bubble Transform: A New Tool for Analysis of Finite Element Methods

Richard S. Falk, Ragnar Winther

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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 Ω⊂Rn into a collection of functions with local support. In fact, for a given simplicial triangulation T of Ω, the associated bubble transform BT produces a decomposition of functions on Ω into a sum of functions with support on the corresponding macroelements. The transform is bounded in both L2 and the Sobolev space H1, 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 cutoff operators.

Original languageEnglish (US)
Pages (from-to)297-328
Number of pages32
JournalFoundations of Computational Mathematics
Issue number1
StatePublished - Feb 1 2016

All Science Journal Classification (ASJC) codes

  • Analysis
  • Computational Mathematics
  • Computational Theory and Mathematics
  • Applied Mathematics


  • Local decomposition of H
  • Preservation of piecewise polynomial spaces
  • Simplicial mesh


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