Inspired by the reduced basis approach and modern numerical multiscale methods, we present a new framework for an efficient treatment of heterogeneous multiscale problems. The new approach is based on the idea of considering heterogeneous multiscale problems as parametrized partial differential equations where the parameters are smooth functions. We then construct, in an offline phase, a suitable localized reduced basis that is used in an online phase to efficiently compute approximations of the multiscale problem by means of a discontinuous Galerkin method on a coarse grid. We present our approach for elliptic multiscale problems and discuss an a posteriori error estimate that can be used in the construction process of the localized reduced basis. Numerical experiments are given to demonstrate the efficiency of the new approach.

@Article{KOH11,
  author       = {Kaulmann, S. and Ohlberger, M. and Haasdonk, B.},
  title        = {A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems},
  journal      = {C. R. Math. Acad. Sci. Paris},
  number       = {23-24},
  volume       = {349},
  pages        = {1233--1238},
  year         = {2011},
  doi          = {10.1016/j.crma.2011.10.024},
  url          = \{/2011/KOH11},
}