Tree-cotree Decomposition for High Order Whitney Finite Elements
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The tree-cotree decomposition is a technique used in the finite element approximation of electromagnetic problems to eliminate the kernel of the curl operator from the discrete curl-conforming space. 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 all-node incidence matrix of a directed graph. This is straightforward for low order finite elements and it can be extended in a very natural way to the high order case using a particular set of degrees of freedom, the so-called weights, that have a clear geometric localization on the mesh. The classical degrees of freedom for high order Whitney finite elements are moments that have not this direct geometrical localization. A particular isomorphism between weights and moments that preserves the matrix of the gradient operator allows to extend this kind of decomposition to the canonical basis for moments.
