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

Adaptive multiscale model reduction for nonlinear parabolic equations using GMsFEM