  In this paper, we present a Constraint Energy Minimizing Generalized
Multiscale Finite Element Method (CEM-GMsFEM) for parabolic equations with
multiscale coefficients, arising from applications in porous media. We will
present the construction of CEM-GMsFEM and rigorously analyze its convergence
for the parabolic equations. The convergence rate is characterized by the
coarse grid size and the eigenvalue decay of local spectral problems, but is
independent of the scale length and contrast of the media. The analysis shows
that the method has a first order convergence rate with respect to coarse grid
size in the energy norm and second order convergence rate with respect to
coarse grid size in $L^2$ norm under some appropriate assumptions. For the
temporal discretization, finite difference techniques are used and the
convergence analysis of full discrete scheme is given. Moreover, a posteriori
error estimator is derived and analyzed. A few numerical results for porous
media applications are presented to confirm the theoretical findings and
demonstrate the performance of the approach.
