, 55:42 | Cite as

A graph approach for the construction of high order divergence-free Raviart–Thomas finite elements

  • A. Alonso RodríguezEmail author
  • J. Camaño
  • E. De Los Santos
  • F. Rapetti


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 \(r \ge 0\), 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. This approach extends to finite elements of high degree the method introduced by Alotto and Perugia (Calcolo 36:233–248, 1999) 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 or the polynomial degree r.


High order Raviart–Thomas finite elements Divergence-free finite elements Spanning tree Oriented graph Incidence matrix 

Mathematics Subject Classification

65N30 05C05 



The second author was partially supported by CONICYT-Chile through Fondecyt project 1180859 and the project AFB170001 of the PIA Program: Concurso Apoyo a Centros Cientificos y Tecnologicos de Excelencia con Financiamiento Basal. The third author was partially supported by a CONICYT (Chile) fellowship.


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Copyright information

© Istituto di Informatica e Telematica del Consiglio Nazionale delle Ricerche 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di TrentoPovo, TrentoItaly
  2. 2.Departamento de Matemática y Física AplicadasUniversidad Católica de la Santísima ConcepciónConcepciónChile
  3. 3.CI2MAUniversidad de ConcepciónConcepciónChile
  4. 4.Departamento de Ingeniería MatemáticaUniversidad de ConcepciónConcepciónChile
  5. 5.Dep. de Mathématiques J.-A. DieudonnéUniv. Côte d’AzurNice cedex 02France

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