A fast algorithm for particle simulations
Journal of Computational Physics
A general convergence result for a functional related to the theory of homogenization
SIAM Journal on Mathematical Analysis
Homogenization and two-scale convergence
SIAM Journal on Mathematical Analysis
A multiscale finite element method for elliptic problems in composite materials and porous media
Journal of Computational Physics
Wavelet-Based Numerical Homogenization
SIAM Journal on Numerical Analysis
The black box multigrid numerical homogenization algorithm
Journal of Computational Physics
Convergence of a Nonconforming Multiscale Finite Element Method
SIAM Journal on Numerical Analysis
Computing
A mixed multiscale finite element method for elliptic problems with oscillating coefficients
Mathematics of Computation
Upscaling methods for a class of convection-diffusion equations with highly oscillating coefficients
Journal of Computational Physics
Iterative multiscale finite-volume method
Journal of Computational Physics
Hi-index | 31.46 |
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.