Exponentially Convergent Multiscale Finite Element Method  

作　　者：Yifan Chen Thomas Y.Hou Yixuan Wang 

机构地区：[1]Applied and Computational Mathematics,Caltech,Pasadena 91106,USA

出　　处：《Communications on Applied Mathematics and Computation》2024年第2期862-878,共17页应用数学与计算数学学报（英文）

基　　金：part supported by the NSF Grants DMS-1912654 and DMS 2205590。

摘　　要：We provide a concise review of the exponentially convergent multiscale finite element method(ExpMsFEM)for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave propagation.The ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions.Unlike most generalizations of the MsFEM in the literature,the ExpMsFEM does not rely on any partition of unity functions.In general,it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov n-width barrier to achieve exponential convergence.Indeed,there are online and offline parts in the function representation provided by the ExpMsFEM.The online part depends on the right-hand side locally and can be computed in parallel efficiently.The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix;they are all independent of the right-hand side,so the stiffness matrix can be used repeatedly in multi-query scenarios.

关 键 词：Multiscale method Exponential convergence Helmholtz's equation Domain decomposition Nonlinear model reduction 

分 类 号：O241.82[理学—计算数学]