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Finite element heterogeneous multiscale method for the wave equation. (English) Zbl 1298.65145

The authors propose a multiscale finite element (FE) method for the numerical homogenization of the wave equation with heterogeneous coefficients. This method differs from the method of H. Owhadi and I. Zhang [Comput. Methods Appl. Mech. Eng. 198, No. 3–4, 397–406 (2008; Zbl 1228.76122)] as it is based on local microcomputations within a macroscopic FE mesh. Therefore, scale separation (e.g., random stationarity) is needed for their method to be effective, but in turn it yields a significant reduction in computational work. Their method also differs from the one of Q. Dong and L. Cao [Appl. Numer. Math. 59, No. 12, 3008–3032 (2009; Zbl 1177.65147)] as it approximates the effective (homogenized) solution; in contrast, however, the computational domain does not need to extend across an integer number of periodic cells; moreover, their oscillatory tensors are not assumed to have any special symmetry and can vary in space nonuniformly; hence, numerical quadrature has to be dealt with in their approach. In contrast to B. Engquist et al.’s [Commun. Math. Sci. 9, No. 1, 33–56 (2011; Zbl 1281.65110)] their method is based on an FE discretization, and they derive fully discrete optimal error estimates in the energy and in the \(L^2\) norm. Such results constitute the first fully discrete error analysis (in space) for the numerical homogenization of the wave equation. Their analysis relies on a new Strang-type lemma for estimating the error obtained by a perturbation of the bilinear form involved in the wave equation. This result allows them to reuse former error estimates for the finite element heterogeneous multiscale method (FE-HMM) for elliptic multiscale equations.

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
74Q10 Homogenization and oscillations in dynamical problems of solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
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