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Preprint 2018-23

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

A graph approach for the construction of high order divergence-free Raviart-Thomas fi nite elements

Abstract:

We 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 ≥ 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. This approach extends to finite elements of high degree a method introduced previously by Alotto and Perugia for finite elements of degree one. The analyzed approach is 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 on the polynomial degree r.

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This preprint gave rise to the following definitive publication(s):

Ana ALONSO-RODRIGUEZ, Jessika CAMAñO, Eduardo DE LOS SANTOS, Francesca RAPETTI: A graph approach for the construction of high order divergence-free Raviart-Thomas fi nite elements. Calcolo, vol. 55, 4, article:42, (2018).

 

 

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