Thesis Summary:

The main goal of this thesis is to extend to the high order case some techniques for the construction of bases of the space of divergence-free Raviart–Thomas finite elements, which are well-known in the case of order one. The knowledge of a basis of the constrained subspace is a convenient alternative to impose the divergence-free condition, avoiding in this way the standard technique that uses Lagrange multipliers. The disadvantage with Lagrange multipliers is that they introduce a new unknown that has to be discretized and leads to a bigger linear system with a matrix that is not symmetric positive definite even though the bilinear form associated to the variational formulation of the problem in the constrained space is symmetric and coercive. Firstly we propose and analyze an efficient algorithm for the construction of a basis of the space of divergence-free Raviart–Thomas finite elements 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 also 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 directed, connected and without self-loop graph. By choosing a spanning tree of this graph, it is possible to identify an invertible square submatrix of the divergence matrix and with this invertible matrix it is easy to compute 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 in cite{AP99} for finite elements of degree one, in other hand, the tree-cotree method can be traced back to Kirchhoff. 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 of the polynomial degree $r$. Next we extend to the high order case a second approach based on the construction of a basis of the range of the curl operator. If the boundary of the domain is connected, the moments of the elements of a high order Raviart–Thomas divergence-free ($RT_{r+1}^0$) basis, are computed by selecting first (in an appropriate way) some elements of a high order Nédélec ($N_{r+1}$) cardinal (dual) basis, and then computing their curls. The method use the graph associated to the gradient operator and a spanning tree of this graph to select the elements of the basis. Finally we apply a divergence-free finite element method to solve a fluid-structure inter- action spectral problem in the three-dimensional case. The main unknowns of the resulting formulation are given by displacements for the fluid and the solid, and the pressure of the fluid on the interface separating the fluid and the solid. The resulting mixed eigenvalue problem is approximated by using appropriate basis of the divergence-free lowest order Raviart–Thomas elements for the fluid, piecewise linear elements for the solid and piece- wise constant elements for the pressure. It is proved that eigenvalues and eigenfunctions are efficiently approximated and some numerical results are presented in order to assess the performance of the method, and also showing that this method avoid spurious modes.

ISI Publications from the Thesis

Ana ALONSO-RODRIGUEZ, Jessika CAMAñO, Eduardo DE LOS SANTOS, Rodolfo RODRíGUEZ: Divergence-free finite elements for the numerical solution of a hydroelastic vibration problem. Numerical Methods for Partial Differential Equations, vol. 39, pp. 163-186, (2023).

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