A dual finite element complex on the barycentric refinement

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 iin \{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.