Direct numerical discretization of stationary and/or transient PDEs with multiscale coefficients requires solution of huge linear systems or very stiff ODEs. The associated complexity can be handled in some cases by use of various multigrid and multilevel solvers.
On the other hand, in the last few years a plethora of multiscale numerical methods based on finite elements have been proposed for various applications. These include the heterogeneous multiscale method, the variational multiscale method, and the multiscale FE (with overlapping variants) as well as subgrid or mortar methods. Most are designed to compute the macroscale average solution, and some are able to recover next order effects. Some methods work best for periodic coefficients, and some can be extended to handle any coefficients including random. Some assume scale separation, and some use special test functions in the classical or mixed variational formulation. Finally, some methods can be naturally applied to transient and nonlinear problems. In the talk we give an overview of main ideas and issues as well as discuss some open problems in various applications.