  In this paper, we develop the constraint energy minimization generalized
multiscale finite element method (CEM-GMsFEM) in mixed formulation applied to
parabolic equations with heterogeneous diffusion coefficients. The construction
of the method is based on two multiscale spaces: pressure multiscale space and
velocity multiscale space. The pressure space is constructed via a set of
well-designed local spectral problems, which can be solved independently. Based
on the computed pressure multiscale space, we will construct the velocity
multiscale space by applying constrained energy minimization. The convergence
of the proposed method is proved.In particular, we prove that the convergence
of the method depends only on the coarse grid size, and is independent of the
heterogeneities and contrast of thediffusion coefficient. Four typical types of
permeability fields are exploited in the numerical simulations, and the results
indicate that our proposed method works well and gives efficient and accurate
numerical solutions.
