Numerical methods for multiscale elliptic problems

We present an overview of the recent development on numerical methods for elliptic problems with multiscale coefficients. We carry out a thorough study of two representative techniques: the heterogeneous multiscale method (HMM) and the multiscale finite element method (MsFEM). For problems with scale separation (but without specific assumptions on the particular form of the coefficients), analytical and numerical results show that HMM gives comparable accuracy as MsFEM, with much less cost. For problems without scale separation, our numerical results suggest that HMM performs at least as well as MsFEM, in terms of accuracy and cost, even though in this case both methods may fail to converge. Since the cost of MsFEM is comparable to that of solving the full fine scale problem, one might expect that it does not need scale separation and still retains some accuracy. We show that this is not the case. Specifically, we give an example showing that if there exists an intermediate scale comparable to H, the size of the macroscale mesh, then MsFEM commits a finite error even with overlapping.