  We introduce a goal-oriented strategy for multiscale computations performed
using the Multiscale Finite Element Method (MsFEM). In a previous work, we have
shown how to use, in the MsFEM framework, the concept of Constitutive Relation
Error (CRE) to obtain a guaranteed and fully computable a posteriori error
estimate in the energy norm (as well as error indicators on various error
sources). Here, the CRE concept is coupled with the solution of an adjoint
problem to control the error and drive an adaptive procedure with respect to a
given output of interest. Furthermore, a local and non-intrusive enrichment
technique is proposed to enhance the accuracy of error bounds. The overall
strategy, which is fully automatic and robust, enables to reach an appropriate
trade-off between certified reliability and computational cost in the MsFEM
context. The performances of the proposed method are investigated on several
illustrative numerical test cases. In particular, the error estimation is
observed to be very accurate, yielding a very efficient adaptive procedure.
