  In this paper, we develop an iterative scheme to construct multiscale basis
functions within the framework of the Constraint Energy Minimizing Generalized
Multiscale Finite Element Method (CEM-GMsFEM) for the mixed formulation. The
iterative procedure starts with the construction of an energy minimizing
snapshot space that can be used for approximating the solution of the model
problem. A spectral decomposition is then performed on the snapshot space to
form global multiscale space. Under this setting, each global multiscale basis
function can be split into a non-decaying and a decaying parts. The
non-decaying part of a global basis is localized and it is fixed during the
iteration. Then, one can approximate the decaying part via a modified
Richardson scheme with an appropriately defined preconditioner. Using this set
of iterative-based multiscale basis functions, first-order convergence with
respect to the coarse mesh size can be shown if sufficiently many times of
iterations with regularization parameter being in an appropriate range are
performed. Numerical results are presented to illustrate the effectiveness and
efficiency of the proposed computational multiscale method.
