## Preprint 2019-28

## Ana Alonso-Rodriguez, Jessika Camaño, Eduardo De Los Santos, Francesca Rapetti:

### A tree-cotree splitting for the construction of divergence-free finite elements: the high order case

### Abstract:

We extend, to Raviart-Thomas finite elements of any degree, two methods for the construction of basis of the space of divergence-free functions that are well established in the case of degree one. The first one computes directly a basis of the kernel of the divergence operator whereas the second one computes a basis of the image of the curl operator that, if the boundary of the domain is not connected, is completed with a basis of the second de Rham cohomology group (namely, the space of divergence-free functions that are not curls). When using the lower order Whitney elements on a tetrahedral mesh, the degrees of freedom are supported on the vertices, edges, faces and tetrahedra of the mesh respectively and, from Stokes theorem, the matrices describing the differential operators gradient, curl and divergence are the transposed of the connectivity matrices of the mesh. This allows the use of a tree-cotree splitting of associated oriented graphs to efficiently construct a basis of either the kernel of the divergence or the image of the curl operator. We prove that these two properties hold true also for r > 0 when using as degrees of freedom a particular realization, based on Berstein polynomials, of the moments. In this work we analyze in detail the second method, the one based on the identification of a basis of the space of the curls of Nédélec finite elements.