Generalized multiscale finite element method for elasticity equations
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Generalized multiscale finite element method for elasticity equations Eric T. Chung · Yalchin Efendiev · Shubin Fu
Received: 20 August 2014 / Accepted: 21 September 2014 / Published online: 5 October 2014 © Springer-Verlag Berlin Heidelberg 2014
Abstract In this paper, we discuss the application of generalized multiscale finite element method (GMsFEM) to elasticity equation in heterogeneous media. We consider steady state elasticity equations though some of our applications are motivated by elastic wave propagation in subsurface where the subsurface properties can be highly heterogeneous and have high contrast. We present the construction of main ingredients for GMsFEM such as the snapshot space and offline spaces. The latter is constructed using local spectral decomposition in the snapshot space. The spectral decomposition is based on the analysis which is provided in the paper. We consider both continuous Galerkin and discontinuous Galerkin coupling of basis functions. Both approaches have their cons and pros. Continuous Galerkin methods allow avoiding penalty parameters though they involve partition of unity functions which can alter the properties of multiscale basis functions. On the other hand, discontinuous Galerkin techniques allow gluing multiscale basis functions without any modifications. Because basis functions are constructed independently from each other, this approach provides an advantage. We discuss the use of oversampling techniques that use snapshots in larger regions to construct the offline space. We provide numerical results to show
This research is partially supported by the Hong Kong RGC General Research Fund (Project Number 400411). E. T. Chung Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, Hong Kong SAR Y. Efendiev (B) · S. Fu Department of Mathematics, Texas A&M University, College Station, TX 77843, USA e-mail: [email protected]; [email protected] Y. Efendiev Numerical Porous Media SRI Center, KAUST, Thuwal, Saudi Arabia
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Int J Geomath (2014) 5:225–254
that one can accurately approximate the solution using reduced number of degrees of freedom. Keywords Multiscale finite element method · Elasticity · Multiscale · Model reduction Mathematics Subject Classification
65N99
1 Introduction Many materials in nature are highly heterogeneous and their properties can vary at different scales. Direct numerical simulations in such multiscale media are prohibitively expensive and some type of model reduction is needed. Multiscale approaches such as homogenization and numerical homogenization (Cao 2005; Abdulle 2006; Schröder 2014; Buck et al. 2013; Francfort and Murat 1986; Oleinik et al. 2009; Vinh and Tung 2011; Liu et al. 2009) have been routinely used to model macroscopic properties and macroscopic behavior of elastic materials. These approaches compute the effective material properties based on representative volume simulations. These properties are further used to solve macroscale equations. In this paper, our goal is to design multiscale
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