  In this paper, we propose a coupled Discrete Empirical Interpolation Method
(DEIM) and Generalized Multiscale Finite element method (GMsFEM) to solve
nonlinear parabolic equations with application to the Allen-Cahn equation. The
Allen-Cahn equation is a model for nonlinear reaction-diffusion process. It is
often used to model interface motion in time, e.g. phase separation in alloys.
The GMsFEM allows solving multiscale problems at a reduced computational cost
by constructing a reduced-order representation of the solution on a coarse
grid. In arXiv:1301.2866, it was shown that the GMsFEM provides a flexible tool
to solve multiscale problems by constructing appropriate snapshot, offline and
online spaces. In this paper, we solve a time dependent problem, where online
enrichment is used. The main contribution is comparing different online
enrichment methods. More specifically, we compare uniform online enrichment and
adaptive methods. We also compare two kinds of adaptive methods. Furthermore,
we use DEIM, a dimension reduction method to reduce the complexity when we
evaluate the nonlinear terms. Our results show that DEIM can approximate the
nonlinear term without significantly increasing the error. Finally, we apply
our proposed method to the Allen Cahn equation.
