Multiscale Finite Element Methods for Elliptic Equations

Here and throughout this chapter, the Einstein convention for repeated indices are assumed. The problem (9.1) a model multiscale problem which arises in the modeling of composite materials and the flow transport in heterogeneous porous media. The main difficulty in solving it by standard finite element method is that when ε is small, the underlying finite element mesh h must be much less than ε which makes the computational costs prohibitive. The multiscale finite element method allows to solve the problem with mesh size h greater than ε.