A Multiscale Finite Element Method for Numerical Homogenization

This paper is concerned with a multiscale finite element method for numerically 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 a fine local mesh, the latter being used for computing independently an adapted finite element basis for the coarse mesh. The main 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 in various examples.