  In this paper, we construct an adaptive multiscale method for solving
H(curl)-elliptic problems in highly heterogeneous media. Our method is based on
the generalized multiscale finite element method. We will first construct a
suitable snapshot space, and a dimensional reduction procedure to identify
important modes of the solution. We next develop and analyze an a posteriori
error indicator, and the corresponding adaptive algorithm. In addition, we will
construct a coupled offline-online adaptive algorithm, which provides an
adaptive strategy to the selection of offline and online basis functions. Our
theory shows that the convergence is robust with respect to the heterogeneities
and contrast of the media. We present several numerical results to illustrate
the performance of our method.
