Adaptive multiscale model reduction for nonlinear parabolic equations using GMsFEM

TL;DR

This paper introduces an adaptive multiscale model reduction technique combining GMsFEM and DEIM to efficiently solve nonlinear parabolic equations like the Allen-Cahn equation, with a focus on online enrichment strategies.

Contribution

It develops and compares adaptive online enrichment methods within GMsFEM and integrates DEIM for nonlinear term approximation in solving multiscale nonlinear parabolic equations.

Findings

DEIM effectively approximates nonlinear terms with minimal error increase.

Adaptive online enrichment improves computational efficiency over uniform methods.

The proposed approach successfully solves the Allen-Cahn equation with reduced computational cost.

Abstract

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.