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

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 not only captures 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 micro scale. Optimal error estimates in the energy norm and the L2-norm are proved over finite time intervals, which imply convergence to the solution from classical homogenization theory when both the macro and the micro scale are refined simultaneously. Numerical experiments illustrate the usefulness of the FE-HMM-L method and corroborate the theory.