The fictitious domain method and applications in wave propagation

Eliane Bécache (1) , Jerónimo Rodríguez Garcia (2, 3) , Chrysoula Tsogka (4)

1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
2 USC - Universidade de Santiago de Compostela [Spain]
3 Departamento de Matemática Aplicada
4 UC Merced - University of California [Merced]

This paper deals with the convergence analysis of the fictitious domain method used for taking into account the Neumann boundary condition on the surface of a crack (or more generally an object) in the context of acoustic and elastic wave propagation. For both types of waves we consider the first order in time formulation of the problem known as mixed velocity-pressure formulation for acoustics and velocity-stress formulation for elastodynamics. The convergence analysis for the discrete problem depends on the mixed finite elements used. We consider here two families of mixed finite elements that are compatible with mass lumping. When using the first one which is less expensive and corresponds to the choice made in a previous paper, it is shown that the fictitious domain method does not always converge. For the second one a theoretical convergence analysis was carried out in [7] for the acoustic case. Here we present numerical results that illustrate the convergence of the method both for acoustic and elastic waves.

Mots clés: fictitious domain method; mixed finite elements; cracks; elastic waves; acoustic waves

Domaines: Mathématiques [math]; Analyse numérique [math.NA]

Eliane Bécache, Jerónimo Rodríguez Garcia, Chrysoula Tsogka. The fictitious domain method and applications in wave propagation. COMPDYN 2009, ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, M. Papadrakakis, N.D. Lagaros, M. Fragiadakis (eds.), 2009. ⟨hal-01853643⟩