In this work we address the Multiscale Spectral Generalized Finite Element Method (MS-GFEM) developed in [I. Babuška and R. Lipton, Multiscale Modeling and Simulation 9 (2011), pp. 373--406]. We outline the numerical implementation of this method and present simulations that demonstrate contrast independent exponential convergence of MS-GFEM solutions. We introduce strategies to reduce the computational cost of generating the optimal oversampled local approximating spaces used here. These strategies retain accuracy while reducing the computational work necessary to generate local bases. Motivated by oversampling we develop a nearly optimal local basis based on a partition of unity on the boundary and the associated A-harmonic extensions.
In this paper, the first large-scale application of multiscale-spectral generalized finite element methods (MS-GFEM) to composite aero-structures is presented. The crucial novelty lies in the introduction of A-harmonicity in the local approximation spaces, which in contrast to [Babuska, Lipton, Multiscale Model. Simul. 9, 2011] is enforced more efficiently via a constraint in the local eigenproblems. This significant modification leads to excellent approximation properties, which turn out to be essential to capture accurately material strains and stresses with a low dimensional approximation s…
Oversampling for the Multiscale Finite Element Method
Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms-Universität Münster（应用与数值数学研究所，明斯特大学）；Institut für Mathematik, Humboldt-Universität zu Berlin（数学研究所，柏林洪堡大学）
Online mixed multiscale finite element method with oversampling and its applications
School of Mathematics and Information Science, Guangzhou University（广州大学数学与信息科学学院）；Department of Mathematics, The Chinese University of Hong Kong（香港中文大学数学系）
In this research, we propose an online basis enrichment strategy within the framework of a recently developed constraint energy minimizing generalized multiscale discontinuous Galerkin method (CEM-GMsDGM). Combining the technique of oversampling, one makes use of the information of the current residuals to adaptively construct basis functions in the online stage to reduce the error of multiscale approximation. A complete analysis of the method is presented, which shows the proposed online enrichment leads to a fast convergence from multiscale approximation to the fine-scale solution. The error…
In this paper, we develop an iterative scheme to construct multiscale basis functions within the framework of the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) for the mixed formulation. The iterative procedure starts with the construction of an energy minimizing snapshot space that can be used for approximating the solution of the model problem. A spectral decomposition is then performed on the snapshot space to form global multiscale space. Under this setting, each global multiscale basis function can be split into a non-decaying and a decaying parts.…
In this paper we present and analyze a constraint energy minimizing generalized multiscale finite element method for convection diffusion equation. To define the multiscale basis functions, we first build an auxiliary multiscale space by solving local spectral problems motivated by analysis. Then constraint energy minimization performed in oversampling domains is exploited to construct the multiscale space. The resulting multiscale basis functions have a good decay property even for high contrast diffusion and convection coefficients. Furthermore, if the number of oversampling layer is chosen …
In this paper, the generalized finite element method (GFEM) for solving second order elliptic equations with rough coefficients is studied. New optimal local approximation spaces for GFEMs based on local eigenvalue problems involving a partition of unity are presented. These new spaces have advantages over those proposed in [I. Babuska and R. Lipton, Multiscale Model.\;\,Simul., 9 (2011), pp.~373--406]. First, in addition to a nearly exponential decay rate of the local approximation errors with respect to the dimensions of the local spaces, the rate of convergence with respect to the size of t…
Optimal Local Approximation Spaces for Generalized Finite Element Methods with Application to Multiscale Problems
ICES and Department of Aerospace Engineering, University of Texas, Austin（德克萨斯大学奥斯汀分校航空航天工程系计算工程研究所）；Department of Mathematics, Louisiana State University（路易斯安那州立大学数学系）
The simulation of the elastodynamics equations at high-frequency suffers from the well known pollution effect. We present a Petrov--Galerkin multiscale sub-grid correction method that remains pollution free in natural resolution and oversampling regimes. This is accomplished by generating corrections to coarse-grid spaces with supports determined by oversampling lengths related to the $\log(k)$, $k$ being the wavenumber. Key to this method are polynomial-in-$k$ bounds for stability constants and related inf-sup constants. To this end, we establish polynomial-in-$k$ bounds for the elastodynamic…
In this research, an online basis enrichment strategy for the constraint energy minimizing generalized multiscale finite element method in mixed formulation is proposed. The online approach is based on the technique of oversampling. One makes use of the information of residual and the data in the partial differential equation such as the source function. The analysis presented shows that the proposed online enrichment leads to a fast convergence from multiscale approximation to the fine-scale solution. The error reduction can be made sufficiently large by suitably selecting oversampling region…
In this paper, we consider the incompressible Stokes flow problem in a perforated domain and employ the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) to solve this problem. The proposed method provides a flexible and systematical approach to construct crucial divergence-free multiscale basis functions for approximating the displacement field. These basis functions are constructed by solving a class of local energy minimization problems over the eigenspaces that contain local information on the heterogeneities. These multiscale basis functions are shown …
In this paper, we consider a pressure-velocity formulation of the heterogeneous wave equation and employ the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) to solve this problem. The proposed method provides a flexible framework to construct crucial multiscale basis functions for approximating the pressure and velocity. These basis functions are constructed by solving a class of local auxiliary optimization problems over the eigenspaces that contain local information on the heterogeneity. Techniques of oversampling are adapted to enhance the computationa…
In this paper, a generalized finite element method (GFEM) with optimal local approximation spaces for solving high-frequency heterogeneous Helmholtz problems is systematically studied. The local spaces are built from selected eigenvectors of carefully designed local eigenvalue problems defined on generalized harmonic spaces. At both continuous and discrete levels, $(i)$ wavenumber explicit and nearly exponential decay rates for local and global approximation errors are obtained without any assumption on the size of subdomains; $(ii)$ a quasi-optimal convergence of the method is established by …
In this work, we propose a local multiscale model reduction approach for the time-domain scalar wave equation in a heterogenous media. A fine mesh is used to capture the heterogeneities of the coefficient field, and the equation is solved globally on a coarse mesh in the discontinuous Galerkin discretization setting. The main idea of the model reduction approach is to extract dominant modes in local spectral problems for representation of important features, construct multiscale basis functions in coarse oversampled regions by constraint energy minimization problems, and perform a Petrov-Galer…
In this paper, we develop and analyze a rigorous multiscale upscaling method for dual continuum model, which serves as a powerful tool in subsurface formation applications. Our proposed method is capable of identifying different continua and capturing non-local transfer and effective properties in the computational domain via constructing localized multiscale basis functions. The construction of the basis functions consists of solving local problems defined on oversampling computational region, subject to the energy minimizing constraints that the mean values of the local solution are zero in …
CEM-GMsFEM for Poisson equations in heterogeneous perforated domains
National Center for Applied Mathematics in Hunan, Xiangtan University（湖南应用数学中心，湘潭大学）；School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）；Department of Mathematics, The Chinese University of Hong Kong（香港中文大学数学系）
