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Computational homology and some applications in electromagnetism
Prof. Ana Alonso Rodriguez, University of Trento
Sala conferenze IMATI-CNR - Martedì 12 Aprile 2016 h.15:00
Abstract. I will present an efficient algorithm for the construction of a basis of the first de Rham cohomology group in a general polyhedral Lipschitz domain $\Omega$ endowed with a tetrahedral mesh. In particular this algorithm constructs a maximal set of independent irrotational vector fields in the space of the Nédélec first order finite elements, that cannot be expressed in $\Omega$ as the gradient of any single-valued scalar potential because they have circulation different from zero along certain cycles of edges of the mesh. An explicit formula for the construction of such a set of discrete vector fields has been provided in [1]. The explicit formula is given in terms of the linking number between some cycles of edges of the mesh and a set of cycles on $\partial \Omega$ that are representatives of a basis of the first homology group of $\mathbb R^3 \setminus \Omega$. However the proposed algorithm is based in a very easy elimination procedure and it uses the explicit formula rarely, only in those cases where it stops without having computed all the degrees of freedom of one of this vector fields.
Working on the dual mesh, an analogous explicit formula and an analogous elimination procedure have been adopted in [2], for the efficient construction of a homological Seifert of a given cycle of edges of the mesh. This algorithm can be used (see [3]) for the computation of a basis of the second relative homology group $H_2(\overline \Omega,\partial \Omega ; \mathbb Z)$.
[1] A. Alonso Rodríguez, E. Bertolazzi, R. Ghiloni and A. Valli.
Construction of a finite element basis of the first de Rham cohomology group and numerical solution of 3D magnetostatic problems. SIAM J. Numer. Anal., 51 (2013), 2380--2402.
[2] A. Alonso Rodríguez, E. Bertolazzi, R. Ghiloni and R. Specogna.
Efficient construction of homological Seifert surfaces. arXiv:1409.5487.
[3] A. Alonso Rodríguez, E. Bertolazzi, R. Ghiloni and R. Specogna.
Construction of geometric bases of $H_2(\overline \Omega,\partial \Omega ; \mathbb Z)$.
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