  In this research, an online basis enrichment strategy for the constraint
energy minimizing generalized multiscale finite element method in mixed
formulation is proposed. The online approach is based on the technique of
oversampling. One makes use of the information of residual and the data in the
partial differential equation such as the source function. The analysis
presented shows that 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. Also, the convergence rate of the enrichment
can be tuned by a user-defined parameter. Numerical results are provided to
illustrate the efficiency of the proposed method.
