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**Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients.**
*(English)*
Zbl 0922.65071

This paper proposes a multiscale finite element method able to solve a class of 2D, second-order elliptic boundary value problems with highly oscillatory coefficients. Typical applications arise in composite materials or flows in porous media. The objective is to develop a numerical method which can capture the effect of small scales on large scales without resolving the small scale details.

It starts by setting the 2D model problem and the associate multiscale finite element method (§2). A review of the homogenization method is developed in §3. The convergence of the approximate method is analyzed when \(h<\varepsilon\) and \(h>\varepsilon\), where \(h\) and \(\varepsilon\) denotes respectively the mesh size and the small scale parameter (§§4 and 5). The study of the asymptotic structure of the discrete linear system is conducted in §6. Finally, some numerical experiments illustrate the reliability of these methods.

This is a very nice paper which addresses important engineering problems and which includes strong mathematical bases as well as validation by pertinent numerical experiments.

It starts by setting the 2D model problem and the associate multiscale finite element method (§2). A review of the homogenization method is developed in §3. The convergence of the approximate method is analyzed when \(h<\varepsilon\) and \(h>\varepsilon\), where \(h\) and \(\varepsilon\) denotes respectively the mesh size and the small scale parameter (§§4 and 5). The study of the asymptotic structure of the discrete linear system is conducted in §6. Finally, some numerical experiments illustrate the reliability of these methods.

This is a very nice paper which addresses important engineering problems and which includes strong mathematical bases as well as validation by pertinent numerical experiments.

Reviewer: M.Bernadou (Le Chesnay)

### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

74S05 | Finite element methods applied to problems in solid mechanics |

76S05 | Flows in porous media; filtration; seepage |

76M10 | Finite element methods applied to problems in fluid mechanics |

65F10 | Iterative numerical methods for linear systems |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |

### Keywords:

homogenization; multiscale finite element method; second-order elliptic boundary value problem; highly oscillatory coefficients; composite materials; flows in porous media; convergence; numerical experiments
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\textit{T. Y. Hou} et al., Math. Comput. 68, No. 227, 913--943 (1999; Zbl 0922.65071)

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### References:

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