  Offline computation is an essential component in most multiscale model
reduction techniques. However, there are multiscale problems in which offline
procedure is insufficient to give accurate representations of solutions, due to
the fact that offline computations are typically performed locally and global
information is missing in these offline information. To tackle this difficulty,
we develop an online local adaptivity technique for local multiscale model
reduction problems. We design new online basis functions within Discontinuous
Galerkin method based on local residuals and some optimally estimates. The
resulting basis functions are able to capture the solution efficiently and
accurately, and are added to the approximation iteratively. Moreover, we show
that the iterative procedure is convergent with a rate independent of physical
scales if the initial space is chosen carefully. Our analysis also gives a
guideline on how to choose the initial space. We present some numerical
examples to show the performance of the proposed method.
