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A dual finite element complex on the barycentric refinement

Authors: Annalisa Buffa and Snorre H. Christiansen
Journal: Math. Comp. 76 (2007), 1743-1769
MSC (2000): Primary 65N30, 65N38
Published electronically: May 3, 2007
MathSciNet review: 2336266
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Abstract: Given a two dimensional oriented surface equipped with a simplicial mesh, the standard lowest order finite element spaces provide a complex $ X^\bullet$ centered on Raviart-Thomas divergence conforming vector fields. It can be seen as a realization of the simplicial cochain complex. We construct a new complex $ Y^\bullet$ of finite element spaces on the barycentric refinement of the mesh which can be seen as a realization of the simplicial chain complex on the original (unrefined) mesh, such that the $ \mathrm{L}^2$ duality is non-degenerate on $ Y^i \times X^{2-i}$ for each $ i\in \{0,1,2\}$. In particular $ Y^1$ is a space of $ \mathrm{curl}$-conforming vector fields which is $ \mathrm{L}^2$ dual to Raviart-Thomas $ \operatorname{div}$-conforming elements. When interpreted in terms of differential forms, these two complexes provide a finite-dimensional analogue of Hodge duality.

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Additional Information

Annalisa Buffa
Affiliation: Istituto di Matematica Applicata e Tecnologie Informatiche - CNR, Via Ferrata 1, 27100 Pavia, Italy

Snorre H. Christiansen
Affiliation: CMA c/o Matematisk Institutt, PB 1053 Blindern, Universitetet i Oslo, NO-0316 Oslo, Norway

Received by editor(s): July 6, 2005
Received by editor(s) in revised form: July 25, 2006
Published electronically: May 3, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.