TY - JOUR

T1 - The Bubble Transform

T2 - A New Tool for Analysis of Finite Element Methods

AU - Falk, Richard S.

AU - Winther, Ragnar

N1 - Funding Information:
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement 339643.
Publisher Copyright:
© 2015, SFoCM.

PY - 2016/2/1

Y1 - 2016/2/1

N2 - 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.

AB - 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.

KW - Local decomposition of H

KW - Preservation of piecewise polynomial spaces

KW - Simplicial mesh

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U2 - 10.1007/s10208-015-9252-1

DO - 10.1007/s10208-015-9252-1

M3 - Article

AN - SCOPUS:84955631527

VL - 16

SP - 297

EP - 328

JO - Foundations of Computational Mathematics

JF - Foundations of Computational Mathematics

SN - 1615-3375

IS - 1

ER -