Error control and adaptivity for heterogeneous multiscale Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. FOR THE MsFEM 1175 Downloaded 11/13/13 to 128.176.171.163. Redistribution subject to SIAM license or copyright; see htt

Abstract. In this work we introduce and analyse a new adaptive Petrov-Galerkin heterogeneous multiscale finite element method (HMM) for monotone elliptic operators with rapid oscillations. In a general heterogeneous setting we prove convergence of the HMM approximations to the solution of a macroscopic limit equation. The major new contribution of this work is an a-posteriori error estimate for the L2-error between the HMM approximation and the solution of the macroscopic limit equation. The a posteriori error estimate is obtained in a general heterogeneous setting with scale separation without assuming pe-riodicity or stochastic ergodicity. The applicability of the method and the usage of the a posteriori error estimate for adaptive local mesh refinement is demonstrated in numerical experiments. The experimental results underline the applicability of the a posteriori error estimate in non-periodic homogeniza-tion settings. 1. Introduction. Thi