论文标题
使用GMSFEM的非线性抛物线方程的自适应多尺度模型降低
Adaptive multiscale model reduction for nonlinear parabolic equations using GMsFEM
论文作者
论文摘要
在本文中，我们提出了一种耦合的离散经验插值法（DEIM）和广义的多尺度有限元方法（GMSFEM），以求解非线性抛物线方程，并应用于Allen-CAHN方程。 Allen-CAHN方程是非线性反应扩散过程的模型。它通常用于及时建模接口运动，例如合金中的相位分离。 GMSFEM允许通过在粗网格上构造解决方案的减少阶表示，以降低的计算成本解决多尺度问题。在Arxiv：1301.2866中，显示GMSFEM通过构建适当的快照，离线和在线空间提供了灵活的工具来解决多尺度问题。在本文中，我们解决了使用在线富集的时间依赖时间的问题。主要贡献是比较不同的在线富集方法。更具体地说，我们比较统一的在线丰富和自适应方法。我们还比较了两种自适应方法。此外，当我们评估非线性项时，我们使用Deim，一种减小维度的方法来降低复杂性。我们的结果表明，Deim可以近似非线性项而不会显着增加误差。最后，我们将提出的方法应用于艾伦·卡恩方程。
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.
