三类多尺度问题的多尺度算法

项目名称： 三类多尺度问题的多尺度算法

项目编号： No.11501399

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 杜锐

作者单位： 苏州大学

项目金额： 18万元

中文摘要： 多尺度问题在科学和工程中普遍存在，通常由带快速振荡系数的微分方程描述。由于该问题的计算复杂性及其对应宏观问题的重要性，针对这类问题的多尺度方法和方法分析都得到了很大的发展。已有的研究主要集中在多尺度问题及其对应宏观问题满足相同形式微分方程的情形。然而很多复杂系统中的问题都依赖于时间，在其对应的宏观问题中体现为关于时间的微分积分方程。本课题的主要目的是研究与此类问题相关的三类多尺度问题的方法及算法分析。第一，研究多孔区域上带振荡系数的Stekloff特征值问题的异质多尺度方法及算法分析；第二，研究一阶含时带振荡系数微分方程的异质多尺度方法及算法分析；第三，研究含时带振荡系数椭圆方程的多尺度有限元方法及算法分析。

中文关键词： 均匀化方法；均匀化理论；有限元方法；收敛性；误差估计

英文摘要： Multiscale problems occur commonly in science and engineering, which are often described by differential equations with highly oscillating coefficients. Due to their complexity and the importance of the corresponding macroscopic problems, many multiscale methods have been proposed and analyzed. Existing studies are mainly on the case where the homogenized equation shares the same form as that of the original equations. However, there are many mutliscale problems arising from complex systems that are time-dependent, which makes homogenized equations integro-differential. The main goal of this project is to study multiscale methods for three classes of multiscale problems that are related to the above mentioned problem. Firstly, the heterogeneous multiscale method for Stekloff eigenvalue problems with highly oscillating coefficients in perforated domains will be studied and analyzed; Secondly, the heterogeneous multiscale method for a class of time-dependent first-order differential equations with highly oscillating coefficients will be studied and analyzed; Thirdly, the multiscale finite element method for a class of time-dependent elliptic problems with highly oscillating coefficients will be studied and analyzed.

英文关键词： homogenization method;homogenization theory;finite element method;convergence;error estimate