A multiscale finite element method for numerical homogenization


			    Gregoire Allaire
		   Centre de Mathematiques Appliquees
		           Ecole Polytechnique
		       91128 PALAISEAU Cedex France
                  Email: gregoire.allaire@polytechnique.fr
             Web page: http://www.cmap.polytechnique.fr/~allaire/


Abstract: 
We are concerned with a multiscale finite element method 
for numericaly solving second order scalar elliptic boundary value
problems with highly oscillating coefficients. In the spirit of previous
other works our method is 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 new idea is the introduction of a composition rule, or
change of variables, for the construction of this finite element basis.
In particular, this allows for a simple treatment of high order finite
element methods. We provide optimal error estimates in the case
of periodically oscillating coefficients. We illustrate our method on
various examples.