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

Abstract

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...

Effective models for the multidimensional wave equation in heterogeneous media over long time and numerical homogenization

“…In the one-dimensional case, we show that Theorem 3.1 reduces to the result obtained in Ref. 4, where a family of such functions is defined in an explicit way. In the multidimensional case, we will give an algorithm to compute the coefficients to obtain an effective solution (the algorithm can be easily modified to obtain other effective solutions).…”

“…As showed in Ref. 4, the coefficients b 0 and a 2 in the effective equation (3.2) can be computed with the solution of one single cell problem. That leads to the explicit parametric definition of a family of effective equations.…”

“…In Ref. 4, the one-dimensional result from Ref. 19 was generalized and using the same technique it was proved that there exists a family of (well-posed) effective equations of the form ( cell problems.…”

“…In this paper, we generalize the result from Ref. 4 to the multi-dimensional case, using the adaptation technique arising from asymptotic development introduced in Ref. 19.…”

“…In this paper (as in Ref. 4 and Ref. 19), the proof of the error estimate is done via the definition of an adaptation operator arising from asymptotic expansion, while in Ref.…”