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A fully discrete a priori analysis of the finite element heterogeneous multiscale method introduced in [A. Abdulle, M. Grote, and C. Stohrer, Multiscale Model. Simul., 12, 2014, pp. 1135--1166] for the wave equation with highly oscillatory coefficients over long time is presented. A sharp a 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 $\epsilon$-periodic coefficients up to time ${\cal O}(\epsilon^{-2})$ in [A. Lamacz, Math. Models Methods Appl. Sci., 21, 2011, pp. 1871--1899]. In this paper we also revisit this result by deriving and analyzing 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.

A Priori Error Analysis of the Finite Element Heterogeneous Multiscale Method for the Wave Equation over Long Time