Finite Element Heterogeneous Multiscale Method for the Wave Equation: Long-Time Effects

Assyr Abdulle (1) , Marcus J. Grote (2) , Christian Stohrer (3, 2)

1 EPFL - Ecole Polytechnique Fédérale de Lausanne
2 Department of Mathematics [Basel]
3 POEMS - Propagation des Ondes : Étude Mathématique et Simulation

A new finite element heterogeneous multiscale method (FE-HMM) is proposed for the numerical solution of the wave equation over long times in a rapidly varying medium. Our new FE-HMM-L method captures not only the short-time behavior of the wave field, well described by classical homogenization theory, but also more subtle long-time dispersive effects, both at a computational cost independent of the microscale. Optimal error estimates in the energy norm and the $L^2$-norm are proved over finite time intervals, which imply convergence to the solution from classical homogenization theory when both the macro- and the microscale are refined simultaneously. Numerical experiments illustrate the usefulness of the FE-HMM-L method and corroborate the theory.

Mots clés: multiscale method; heterogeneous media; numerical homogenization; wave equa- tion; second-order hyperbolic problems; long-time behavior; dispersive waves

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Assyr Abdulle, Marcus J. Grote, Christian Stohrer. Finite Element Heterogeneous Multiscale Method for the Wave Equation: Long-Time Effects. Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 2014, 12 (3), pp.1230-1257. ⟨10.1137/13094195X⟩. ⟨hal-01111169⟩