  In this paper, we develop an online basis enrichment method with the mortar
mixed finite element method, using the oversampling technique, to solve for
flow problems in highly heterogeneous media. We first compute a coarse grid
solution with a certain number of offline basis functions per edge, which are
chosen as standard polynomials basis functions. We then iteratively enrich the
multiscale solution space with online multiscale basis functions computed by
using residuals. The iterative solution converges to the fine scale solution
rapidly. We also propose an oversampling online method to achieve faster
convergence speed. The oversampling refers to using larger local regions in
computing the online multiscale basis functions. We present extensive numerical
experiments(including both 2D and 3D) to demonstrate the performance of our
methods for both steady state flow, and two-phase flow and transport problems.
In particular, for the time dependent two-phase flow and transport problems, we
apply the online method to the initial model, without updating basis along the
time evolution. Our numerical results demonstrate that by using a few number of
online basis functions, one can achieve a fast convergence.
