A family of effective equations that capture the long time dispersive effects of wave propagation in heterogeneous media in an arbitrary large periodic spatial domain Ω ⊂ R d over long time is proposed and analyzed. For a wave equation with highly oscillatory periodic spatial tensors of characteristic length ε, we prove that the solution of any member of our family of effective equations are ε-close in the L ∞ (0, T ε , L 2 (Ω)) norm to the true oscillatory wave over a time interval of length T ε = O(ε −2 ). We show that the previously derived effective equation in [Dohnal, Lamacz, Schweizer, Multiscale Model. Simul., 2014] belongs to our family of effective equation. Moreover, while Bloch waves techniques were previously used, we show that asymptotic expansion techniques give an alternative way to derive such effective equations. An algorithm to compute the tensors involved in the dispersive equation and allowing for efficient numerical homogenization methods over long time is proposed.
Abstract. A fully discrete a priori analysis of the finite element heterogenenous multiscale method (FE-HMM) introduced in [A. Abdulle, M. Grote, C. Stohrer, Multiscale Model. Simul. 2014] 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 ε-periodic coefficients up to time O(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. 1. Introduction. The wave equation in heterogenous media is used in a number of scientific and engineering applications such as seismic inversion, medical imaging or the manufacture of composite materials. When the typical size of the heterogeneities (denoted here by ε) is much smaller than the scale of interest, standard numerical methods such as the finite difference method (FDM) or the finite element method (FEM) become prohibitively expensive as scale resolution is needed for the mesh sizes. In such situations, homogenization theory (see [12,25,17,29]) provides a systematic procedure to derive an effective equation for the highly oscillatory wave equation, whose solution no longer oscillates on the ε-scale (see [13] for the specific case of the wave equation).However in practice, no explicit solutions for the effective equation is available (usually obtained by the so-called G-limit of a sequence of differential operators [13]) hence multiscale numerical methods are required. The method considered in this paper is based on the heterogeneous multiscale methods (HMM) [21,3,4]. In this framework, a macroscopic effective equation is computed on a macroscopic grid that does not resolve the fine scale oscillation. The data of the effective equation are recovered "on the fly" by solving micro problems on sampling domains with size proportional to ε, hence at a cost independent of ε. A finite difference scheme based on the HMM (FD-HMM) was proposed by Engquist, Holst and Runborg [22] and a finite element heterogeneous multiscale method (FE-HMM) was later proposed in [5] together with a fully discrete analysis of the method. We mention also upscaling methods that do not rely on scale separation [33,30,14] but on coarse multiscale basis functions obtained by solving local problems on each macro element of the computational domain. In contrast to homogenized based methods the computational cost to obtain the macro model is no longer independent of ε.Classical homogenization describes well the propagation of waves in a strongly heterogeneous medium for short time. The true oscillatory solution however deviates from the classical homogenizat...
Abstract. A family of effective equations for the wave equation in locally periodic media over long time is derived. In particular, explicit formulas for the effective tensors are provided. To validate the derivation, an a priori error estimate between the effective solutions and the original wave is proved. As the dependence of the estimate on the domain is explicit, the result holds in arbitrarily large periodic hypercube. This constitutes the first analysis for the description of long time effects for the wave equation in locally periodic media. Thanks to this result, the long time a priori error analysis of the numerical homogenization method presented in [A. Abdulle and T. Pouchon, SIAM J. Numer. Anal., 54, 2016Anal., 54, , pp. 1507Anal., 54, -1534 is generalized to the case of a locally periodic tensor.
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