A dual finite element complex on the barycentric refinement

Abstract

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 Y2i is in L2 duality with Xi. In terms of differential forms this provides a finite element analogue of Hodge duality.