  In this research, we propose an online basis enrichment strategy within the
framework of a recently developed constraint energy minimizing generalized
multiscale discontinuous Galerkin method (CEM-GMsDGM). Combining the technique
of oversampling, one makes use of the information of the current residuals to
adaptively construct basis functions in the online stage to reduce the error of
multiscale approximation. A complete analysis of the method is presented, which
shows the proposed online enrichment leads to a fast convergence from
multiscale approximation to the fine-scale solution. The error reduction can be
made sufficiently large by suitably selecting oversampling regions and the
number of oversampling layers. Further, the convergence rate of the enrichment
algorithm depends on a factor of exponential decay regarding the number of
oversampling layers and a user-defined parameter. Numerical results are
provided to demonstrate the effectiveness and efficiency of the proposed online
adaptive algorithm.
