Localized harmonic characteristic basis functions for multiscale finite element methods
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Localized harmonic characteristic basis functions for multiscale finite element methods Leonardo A. Poveda1 · Juan Galvis2 · Victor M. Calo3,4
Received: 29 October 2016 / Revised: 29 January 2017 / Accepted: 23 February 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
Abstract We solve elliptic systems of equations posed on highly heterogeneous materials. Examples of this class of problems are composite structures and geological processes. We focus on a model problem which is a second-order elliptic equation with discontinuous coefficients. These coefficients represent the conductivity of a composite material. We assume a background with a low conductivity that contains inclusions with different thermal properties. Under this scenario, we design a multiscale finite element method to efficiently approximate solutions. The method is based on an asymptotic expansion of the solution in terms of the ratio between the conductivities. The resulting method constructs (locally) finite element basis functions (one for each inclusion). These bases generate the multiscale finite element space where the approximation of the solution is computed. Numerical experiments show the good performance of the proposed methodology. Keywords Elliptic equation · Asymptotic expansions · High-contrast coefficients · Multiscale finite element method · Harmonic characteristic function
Communicated by Armin Iske.
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Leonardo A. Poveda [email protected] Juan Galvis [email protected] Victor M. Calo [email protected]
1
Departamento de Matemática Aplicada, Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, Brazil
2
Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá D.C., Colombia
3
Applied Geology Department, Western Australian School of Mines, Curtin University, Western Australia, WA 6102, Australia
4
Mineral Resources, Commonwealth Scientific and Industrial Research Organisation (CSIRO), Kensington, WA 6152, Australia
123
L. A. Poveda et al.
Mathematics Subject Classification 35Q35 · 65M60 · 35J50
1 Introduction Many physical and engineering applications naturally require multiscale solutions. This is especially true for problems related to metamaterials, composite materials, and porous media flows; see Chen and Lipton (2013), Berlyand and Novikov (2002), Cao et al. (2003), Li (2011), Ozgun and Kuzuoglu (2013), Epov et al. (2015), Zhou et al. (2012). The mathematical and numerical analysis for these problems are challenging since they are governed by elliptic equations with high-contrast coefficients (Hou and Wu 1997; Chen and Hou 2013; Ming and Yue 2006; Galvis and Efendiev 2010; Calo et al. 2014). For instance, in the modeling of composite materials, their conducting or elastic properties are modeled by discontinuous coefficients. The value of the coefficient can vary several orders of magnitude across discontinuities. Problems with these jumps are referred to as high-contrast problems. Similarly, the coefficient is denoted as a high-contra
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