## Section: New Results

### Construction of divergence-free bases

Participants : Francesca Rapetti, Ana Alonso Rodriguez.

I have worked to propose and analyze an efficient algorithm for the computation of a basis of the space of divergence-free Raviart-Thomas finite elements. The algorithm is based on graph techniques. The key point is to realize that, with very natural degrees of freedom for fields in the space of Raviart-Thomas finite elements of degree r + 1 and for elements of the space of discontinuous piecewise polynomial functions of degree $r\ge 0$, the matrix associated with the divergence operator is the incidence matrix of a particular graph. By choosing a spanning tree of this graph, it is possible to identify an invertible square submatrix of the divergence matrix and to compute easily the moments of a field in the space of Raviart-Thomas finite elements with assigned divergence. The analyzed approach is then used to construct a basis of the space of divergence-free Raviart-Thomas finite elements. The numerical tests show that the performance of the algorithm depends neither on the topology of the domain nor or the polynomial degree r [16].