Small trees for high order whitney elements

In applied computations the need arises to define, for example, a discrete field with assigned curl or to represent a div-free field in a given discrete space. In the low degree case this need is often fulfilled by employing tree and co-tree techniques. The definition of tree and co-tree is thus revisited here in the frame of high order Whitney element reconstructions. We consider the case of fields that are reconstructed in a contractible polyhedral domain Ω∈ ℝ3, with connected boundary ∂Ω, starting from their weights over suitable “small simplices” in a simplicial mesh ℳ of the domain Ω̄.