  This paper investigates an efficient exponential integrator generalized
multiscale finite element method for solving a class of time-evolving partial
differential equations in bounded domains. The proposed method first performs
the spatial discretization of the model problem using constraint energy
minimizing generalized multiscale finite element method (CEM-GMsFEM). This
approach consists of two stages. First, the auxiliary space is constructed by
solving local spectral problems, where the basis functions corresponding to
small eigenvalues are captured. The multiscale basis functions are obtained in
the second stage using the auxiliary space by solving local energy minimization
problems over the oversampling domains. The basis functions have exponential
decay outside the corresponding local oversampling regions. We shall consider
the first and second-order explicit exponential Runge-Kutta approach for
temporal discretization and to build a fully discrete numerical solution. The
exponential integration strategy for the time variable allows us to take full
advantage of the CEM-GMsFEM as it enables larger time steps due to its
stability properties. We derive the error estimates in the energy norm under
the regularity assumption. Finally, we will provide some numerical experiments
to sustain the efficiency of the proposed method.
