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A priori error analysis of the finite element heterogeneous multiscale method for the wave equation over long time. (English) Zbl 1382.65303

Summary: A fully discrete a priori analysis of the finite element heterogeneous multiscale method introduced in [the first author et al., Multiscale Model. Simul. 12, No. 3, 1230–1257 (2014; Zbl 1315.65084)] 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 \(\varepsilon\)-periodic coefficients up to time \(\mathcal O(\varepsilon^{-2})\) in [A. Lamacz, Math. Models Methods Appl. Sci. 21, No. 9, 1871–1899 (2011; Zbl 1252.35067)]. 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.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35B10 Periodic solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
35L05 Wave equation
74Q10 Homogenization and oscillations in dynamical problems of solid mechanics
74Q15 Effective constitutive equations in solid mechanics
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