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Given a two dimensional oriented surface equipped with a simplicial mesh, the standard lowest order finite element spaces provide a complex X* 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* 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 L 2 duality is non-degenerate on Y i × X 2-i for each i ∈ {0,1,2}. In particular Y 1 is a space of curl-conforming vector fields which is L 2 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.

A dual finite element complex on the barycentric refinement