Multiscale finite element methods

Grégoire Allaire

This course will describe some multiscale finite element methods for numerically solving second order scalar elliptic boundary value problems with highly oscillating coefficients. These methods are based on the coupling of a coarse global mesh and of a fine local mesh, the latter one being used for computing independently an adapted finite element basis for the coarse mesh. The main point is that the coarse mesh size can be much larger than the heterogeneities characteristic length. Optimal error estimates can be obtained in the case of periodically oscillating coefficients. However these methods are not restricted in practice to the periodic setting.

• [1] G. ALLAIRE, R. BRIZZI, A multiscale finite element method for numerical homogenization, submitted, Internal report, n. 545, CMAP, Ecole Polytechnique (2004).
• [2] T. ARBOGAST, Numerical subgrid upscaling of two-phase flow in porous media, in Numerical treatment of multiphase flows in porous media, Lecture Notes in Physics, vol. 552, Chen, Ewing and Shi eds., pp.35-49 (2000).
• [3] T. Y. HOU, X.-H. WU, A multiscale finite element method for elliptic problems in composite materials and porous media, Journal of computational physics 134, 169-189, (1997).
• [4] A.-M. MATACHE, I. BABUSKA, C. SCHWAB, Generalized p-FEM in homogenization, Numer. Math. 86, 319--375 (2000).