A dual finite element complex on the barycentric refinement

A simplicial mesh on an oriented two-dimensional surface gives rise to a complex $X^{?}$ of finite element spaces centered on divergence conforming Raviart–Thomas vector fields and naturally isomorphic to the simplicial cochain complex. On the barycentric refinement of such a mesh, we construct finite element spaces forming a complex $Y^{?}$, centered around curl conforming vector fields, naturally isomorphic to the simplicial chain complex on the original mesh and such that $Y^{2?i}$ is in ${\mathrm{L}}^2$ duality with $X^i$. In terms of differential forms this provides a finite element analogue of Hodge duality. To cite this article: A. Buffa, S.H. Christiansen, C. R. Acad. Sci. Paris, Ser. I 340 (2005).