A dual finite element complex on the barycentric refinement

Source abstract

Given a two dimensional oriented surface equipped with a simplicial mesh, the standard lowest order finite element spaces provide a complex X ∙ 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 ∙ 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 L 2 \mathrm {L}^2 duality is non-degenerate on Y i × X 2 − i Y^i \times X^{2-i} for each i ∈ { 0 , 1 , 2 } i\in \{0,1,2\} . In particular Y 1 Y^1 is a space of c u r l \mathrm {curl} -conforming vector fields which is L 2 \mathrm {L}^2 dual to Raviart-Thomas div \operatorname {div} -conforming elements. When interpreted in terms of differential forms, these two complexes provide a finite-dimensional analogue of Hodge duality.