Construction of a spanning tree for high-order edge elements

The well-known tree–cotree gauging method for low-order edge finite elements is extended to high-order approximations, within the first family of Nédélec finite element spaces. The key point in this method is the identification of degrees of freedom for edge and nodal finite element spaces, such that the matrix of the gradient operator is the transpose of the all nodes incidence matrix of a directed graph. This is straightforward for low-order finite elements, and it can be proved that it is still possible in the high-order case using either moments or weights as degrees of freedom. In the case of weights, a geometrical realization of the graph associated with the gradient operator is very natural. We recall in details the definition of weights and present an algorithm for the construction of a spanning tree of this graph. The starting point of the algorithm is a spanning tree of the graph given by vertices and edges of the mesh (the so-called global spanning tree, that is the one used in the low-order case). This global step, interpreted in the high-order sense, is enriched locally, with a loop over the elements of the mesh, with arcs corresponding to edge, face, and volume degrees of freedom required in the high-order case.