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# A Dual Finite Element Complex on the Barycentric Refinement

Annalisa Buffa and Snorre H. Christiansen
Mathematics of Computation
Vol. 76, No. 260 (Oct., 2007), pp. 1743-1769
Stable URL: http://www.jstor.org/stable/40234460
Page Count: 27
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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 $X^e$ 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 $y^e$ 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 L² duality is non-degenerate on $y^i \times X^{2 - i}$ for each i ∈ {0,1,2}. In particular Y¹ is a space of curl-conforming vector fields which is L² dual to Raviart-Thomas 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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