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This paper introduces an framework for adaptivity for a class of heterogeneous multiscale finite element methods for elliptic problems, which is suitable for  a posteriori  error estimation with separated quantification of the model error as well as the macroscopic and microscopic discretization errors. The method is derived within a general framework for ‘goal-oriented’ adaptivity, the so-called Dual Weighted Residual (DWR) method. This allows for a systematic  a posteriori  balancing of multiscale modeling and discretization. The developed method is tested numerically at elliptic diffusion problems for different types of heterogeneous oscillatory coefficients.

Duality-based adaptivity in finite element discretization of heterogeneous multiscale problems