A mixed multiscale finite element method for elliptic problems with oscillating coefficients.(English)Zbl 1017.65088

This paper is motivated by the numerical simulation of flow transport in highly heterogeneous porous media. By using the most elaborated tools of mathematical finite element methods, the authors analyze a mixed multiscale finite element method with an over-sampling technique which is very well suited for solving second-order elliptic equations with rapidly oscillating coefficients. The multiscale finite element bases are constructed by locally solving Neumann boundary value problems.
By assuming that the oscillating coefficients are locally periodic and by using homogenization techniques, the authors prove the convergence of the method. Finally, they report some numerical experiments which demonstrate the efficiency and accuracy of the proposed method.
This paper is nicely written and looks like a strong mathematical and. numerical contribution to the subject.

MSC:

 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35J25 Boundary value problems for second-order elliptic equations 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 76M10 Finite element methods applied to problems in fluid mechanics 76M50 Homogenization applied to problems in fluid mechanics 76S05 Flows in porous media; filtration; seepage
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References:

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