基于降基多尺度有限元的PGD方法及其在含参数椭圆方程中的应用  

PROPER GENERALIZED DECOMPOSITION METHOD BASED ON REDUCED MULTISCALE FINITE ELEMENT AND ITS APPLICATIONS FOR THE PARAMETRIZED ELLIPTIC EQUATIONS

作　　者：周宇[1] 李秋齐 Zhou Yu Li Qiuqi(College of Mathematics and Econometrics, Hunan University, Changsha, 10082, China)

机构地区：[1]湖南大学数学与计量经济学院,长沙410082

出　　处：《数值计算与计算机应用》2017年第2期105-122,共18页Journal on Numerical Methods and Computer Applications

基　　金：国家自然科学基金项目(11471107)资助项目

摘　　要：为了提高模拟多尺度模型的效率,提出基于降基多尺度有限元的广义特征分解方法.广义多尺度有限元方法是模拟多尺度模型的一种有效方法,在粗网格上构造局部基函数,不仅反映了细尺度上的信息,而且能减少大量的计算量.在广义多尺度有限元方法的框架下,通过交叉验证的思想将多尺度模型映射到降基多尺度有限元空间上,提出基于交叉验证的降基多尺度有限元方法.最后,结合广义特征分解方法和基于交叉验证的降基多尺度有限元方法,将其应用于带参数椭圆偏微分方程的计算.数值例子表明,广义特征分解方法和基于交叉验证的降基多尺度有限元方法相结合,不仅比广义多尺度有限元方法具有更高精度,而且能提高在线计算效率.In order to efficiently approximate the multiscale model, proper generalized decomposition （PGD） based on reduced multiscale finite element method is introduced. Generalized multiscale finite element method （GMsFEM） is an effective method to approximate multi- scale model, the multiple scales can be captured by constructing the multiscale basis in the coarse grid instead of solving the model in a very fine mesh. In the framework of GMsFEM, cross-validation is applied to identify the optimal generalized multiscale finite element basis, thus a reduced order multiscale model is obtained by projecting the original full order model onto the reduced multiscale finite element space. For each iteration of PGD, reduced order rnultiscale model is used to approximate the spatial space, which is the main idea of proper generalized decomposition （PGD） based on reduced multiscale finite element method. To illustrate the efficacy of the proposed methods, a few numerical examples for elliptic PDEs with multsicale and random inputs are presented.

关 键 词：广义特征分解 广义多尺度有限元方法 贪婪取样方法 交叉验证 降基方法 

分 类 号：O241.82[理学—计算数学]