MATHICSE Technical Report : A priori error analysis of the finite element heterogenenous multiscale method for the wave equation in heterogenenous media over long time

A fully discrete a priori analysis of the finite element heterogenenous multiscale method (FE-HMM) introduced in [A. Abdulle, M. Grote, C. Stohrer, MultiscaleModel. Simul. 2014] for the wave equation with highly oscillatory coefficients over long time is presented. A sharpa priori convergence rate for the numerical method is derived for long time intervals. The effective model over long time is a Boussinesq-type equation that has been shown to approximate the one-dimensional multiscale wave equation with ε-periodic coefficients up to timeO(ε−2) in [Lamacz, Math. Models Methods Appl. Sci., 2011]. In this paper we also revisit this result by deriving and analysing a family of effective Boussinesq-type equations for the approximation of the multiscale wave equation that depends on the normalization chosen for certain micro functions used to define the macroscopic models.