Finite Element Approximation of the Neumann Eigenvalue Problem in Domains with Multiple Cracks

Zakaria Belhachmi (1) , Dorin Bucur (2) , Jean-Marc Sac-Epee (3)

1 LMAM - Laboratoire de Mathématiques et Applications de Metz
2 LAMA - Laboratoire de Mathématiques
3 IECL - Institut Élie Cartan de Lorraine

We study the Neumann-Laplacian eigenvalue problem in domains with multiple cracks. We derive a mixed variational formulation which holds on the whole geometric domain (including the cracks) and implement efficient finite element discretizations for the computation of eigenvalues. Optimal error estimates are given and several numerical examples are presented, confirming the efficiency of the method. As applications, we numerically investigate the behavior of the low eigenvalues in domains with a high number of cracks.

Domaines: Mathématiques [math]

Zakaria Belhachmi, Dorin Bucur, Jean-Marc Sac-Epee. Finite Element Approximation of the Neumann Eigenvalue Problem in Domains with Multiple Cracks. IMA Journal of Numerical Analysis, 2006, 86 (3), pp.171--184. ⟨10.1093/imanum/drl002⟩. ⟨hal-02893290⟩