In this work we introduce and analyse a heterogeneous multiscale finite element method (HMM) for monotone elliptic operators with rapid oscillations. We first present a macroscopic limit problem for the oscillating non-linear equations and then prove the convergence of the HMM approximations to the solution of the macroscopic limit equation. On the basis of this identification, we derive an a-posteriori error estimate with duality techniques. The general applicability of the method and its corresponding error estimate is demonstrated in numerical experiments. In particular we state and examine two strategies for adaptive mesh refinement based on the error estimate.

@TechReport{HO11,
  author       = {Henning, P. and Ohlberger, M.},
  title        = {A-posteriori error estimation for a heterogeneous multiscale method for monotone operators and beyond a periodic setting},
  institution  = {FB 10 , Universit\ät M\ünster},
  number       = {01/11 - N},
  month        = {march},
  year         = {2011},
  type         = {Preprint},
  keywords     = {multiscale method, monotone operators, a-posteriori error estimate},
  url          = \{/2011/HO11},
}