Finite Element Heterogeneous Multiscale Methods with Near Optimal Computational Complexity

This paper is concerned with a numerical method for multiscale elliptic problems. Using the framework of the Heterogeneous Multiscale Methods (HMM), we propose a micro-macro approache which combines finite element method (FEM) for the macroscopic solver and the pseudospectral method for the micro solver. Unlike the micro-macro methods based on standard FEM proposed so far in HMM we obtain, for periodic homogenization problems, a method that has almostlinear complexity in the number of degrees of freedom of the discretization of the macro (slow) variable.