Numerical methods for multiscale elliptic problems
This article presents an overview of the recent development on the numerical methods for elliptic problems with multiscale coefficients of the form \[ -\text{div}(a^{\varepsilon}(x)\nabla u^{\varepsilon}(x)) = f(x), \quad D, \] subject to the boundary condition \[ u^{\varepsilon}(x) = g(x), \quad \partial D, \] where \(\varepsilon \ll 1\) is a parameter that represents the ratio of the smallest and largest scales in the problem. The authors have carried out a thorough study of two representative techniques, namely, the heterogeneous multiscale method (HMM), and the multiscale finite element method (MsFEM). An important issue is how the cost of these methods compares with techniques such as multigrid for solving the full fine scale problem. In particular, whether some special features of the problems, such as scale separation, can be exploited to save cost. Nearly all the methods reviewed do have such savings for the special problem of periodic homogenization. For more general problems, the framework of HMM still allows to take full advantage of any scale separation in the problem and therefore reduces the cost. This is difficult from the MsFEM, which in general incurs a cost that is comparable to that of solving the full scale problem. 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, the 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. It is shown that this is not the case. Specifically, the authors give an example showing that if there exists an intermediate scale comparable to \(H\), the step size of the macroscale mesh, then MsFEM commits a finite error even with overlapping.
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