Numerical approximation schemes for multi-dimensional wave equations in asymmetric spaces

Vincent Lescarret (1) , Enrique Zuazua (2)

1 L2S - Laboratoire des signaux et systèmes
2 Departamento de Matemáticas [Madrid]

We develop finite difference numerical schemes for a model arising in multi-body structures, previously analyzed by H. Koch and E. Zuazua [13], constituted by two n-dimensional wave equations coupled with a (n − 1)-dimensional one along a flexible interface. That model, under suitable assumptions on the speed of propagation in each media, is well-posed in asymmetric spaces in which the regularity of solutions differs by one derivative from one medium to the other. Here we consider a flat interface and analyze this property at a discrete level, for finite difference and mixed finite element methods on regular meshes parallel to the interface. We prove that those methods are well-posed in such asymmetric spaces uniformly with respect to the mesh-size parameters and we prove the convergence of the numerical solutions towards the continuous ones in these spaces. In other words, these numerical methods that are well-behaved in standard energy spaces, preserve the convergence properties in these asymmetric spaces too. These results are illustrated by several numerical experiments.

Domaines: Physique mathématique [math-ph]

Vincent Lescarret, Enrique Zuazua. Numerical approximation schemes for multi-dimensional wave equations in asymmetric spaces. Mathematics of Computation, 2015, Mathematics of computations, 84, pp.119-152. ⟨10.1090/S0025-5718-2014-02887-1⟩. ⟨hal-01322032⟩