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44
Generalized Multiscale Finite Element Methods (GMsFEM)
, 2013
"... In this paper, we propose a general approach called Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations for problems without scale separation over a complex input space. As in multiscale finite element methods (MsFEMs), the main idea of the proposed approach i ..."
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Cited by 30 (10 self)
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In this paper, we propose a general approach called Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations for problems without scale separation over a complex input space. As in multiscale finite element methods (MsFEMs), the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space. In the proposed approach, we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage. The enrichment in the online stage is performed based on a spectral decomposition of the offline space. In the online stage, for any input parameter, a multiscale space is constructed to solve the global problem on a coarse grid. The online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues. The computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be reused for solving the forward problem with any forcing and boundary condition. Compared with the other approaches where global snapshots are used, the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarsegrid level. We present various examples in the paper and some numerical results to demonstrate the effectiveness of our method. 1
Analysis of a twolevel Schwarz method with coarse spaces based on local DirichlettoNeumann maps
 Comput. Methods Appl. Math
"... Schwarz method with coarse spaces based on local Dirichlet–to–Neumann maps. computer ..."
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Cited by 15 (7 self)
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Schwarz method with coarse spaces based on local Dirichlet–to–Neumann maps. computer
Analysis of FETI methods for multiscale PDEs  Part II: Interface variation
, 2009
"... In this article we give a new rigorous condition number estimate of the finite element tearing and interconnecting (FETI) method and a variant thereof, allfloating FETI. We consider the scalar elliptic equation in a two or threedimensional domain with a highly heterogeneous (multiscale) diffusi ..."
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Cited by 12 (7 self)
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In this article we give a new rigorous condition number estimate of the finite element tearing and interconnecting (FETI) method and a variant thereof, allfloating FETI. We consider the scalar elliptic equation in a two or threedimensional domain with a highly heterogeneous (multiscale) diffusion coefficient. This coefficient is allowed to have large jumps not only across but also along subdomain interfaces and in the interior of the subdomains. In other words, the subdomain partitioning does not need to resolve any jumps in the coefficient. Under suitable assumptions, we can show that the condition numbers of the onelevel and the allfloating FETI system are robust with respect to strong variations in the contrast in the coefficient. We get only a dependence on some geometric parameters associated with the coefficient variation. In particular, we can show robustness for socalled face, edge, and vertex islands in highcontrast media. As a central tool we prove and use new weighted Poincaré and discrete Sobolev type inequalities that are explicit in the weight. Our theoretical findings are confirmed in a series of numerical experiments.
Scaling Up through Domain Decomposition
, 2009
"... In this paper we discuss domain decomposition parallel iterative solvers for highly heterogeneous problems of flow and transport in porous media. We are particularly interested in highly unstructured coefficient variation where standard periodic or stochastic homogenisation theory is not applicable. ..."
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Cited by 6 (3 self)
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In this paper we discuss domain decomposition parallel iterative solvers for highly heterogeneous problems of flow and transport in porous media. We are particularly interested in highly unstructured coefficient variation where standard periodic or stochastic homogenisation theory is not applicable. When the smallest scale at which the coefficient varies is very small it is often necessary to scale up the equation to a coarser grid to make the problem computationally feasible. Standard upscaling or multiscale techniques, require the solution of local problems in each coarse element, leading to a computational complexity that is at least linear in the global number N of unknowns on the subgrid. Moreover, except for the periodic and the isotropic random case, a theoretical analysis of the accuracy of the upscaled solution is not yet available. Multilevel iterative methods for the original problem on the subgrid, such as multigrid or domain decomposition, lead to similar computational complexity (i.e. O(N)) and are therefore a viable alternative. However, previously no theory was available guaranteeing the robustness of these methods to large coefficient variation. We review a sequence of recent papers where simple variants of domain decomposition methods, such as overlapping Schwarz and onelevel FETI, are proposed that are robust to strong coefficient variation. Moreover, we extend the theoretical results for the first time also to the dualprimal variant of FETI.
P.: Multiscale spectral AMGe solvers for highcontrast flow problems
, 2012
"... Abstract. We construct and analyze multigrid methods with nested coarse spaces for secondorder elliptic problems with highcontrast multiscale coecients. The design of the methods utilizes stable multilevel decompositions with a bound that generally grows with the number of levels. To stabilize thi ..."
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Abstract. We construct and analyze multigrid methods with nested coarse spaces for secondorder elliptic problems with highcontrast multiscale coecients. The design of the methods utilizes stable multilevel decompositions with a bound that generally grows with the number of levels. To stabilize this growth, in our theory, we use AMLIcycle multigrid which leads to an overall optimal cost algorithm. The robustness, with respect to the contrast, is guaranteed due to the combined eect of the Schwarz smoothers used and the spectral construction of the coarse bases. More specically, in order to obtain an optimal multilevel decomposition, we combine multigrid ideas in the recent twolevel methods in [Multiscale Model. Simul. 8(5), 16211644], and earlier, in the elementbased algebraic multigrid methods (or AMGe), that use local spectral problems to enrich the coarse space. In general, the intermediate coarse spaces need to be enriched in order to get contrastindependent convergence. The general techniques presented here allow us to study the problem of an optimal enrichment in the sense of enriching with a minimal number of extra coarse degrees of freedom. Thus, the methods we develop are optimal, with respect to both the contrast and the number of levels used. Moreover, we have the potential to achieve this goal with a minimal number of coarse degrees of freedom. We present numerical results that illustrate our theoretical ndings. 1.
ROBUST MULTILEVEL METHODS FOR GENERAL SYMMETRIC POSITIVE DEFINITE OPERATORS ∗
"... Abstract. An abstract robust multilevel method for solving symmetric positive definite systems resulting from discretizing elliptic partial differential equations is developed. The term “robust” refers to the convergence rate of the method being independent of discretization parameters, i.e., the pr ..."
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Abstract. An abstract robust multilevel method for solving symmetric positive definite systems resulting from discretizing elliptic partial differential equations is developed. The term “robust” refers to the convergence rate of the method being independent of discretization parameters, i.e., the problem size, and problem parameters. Important instances of such problem parameters are in particular (highly varying) coefficients. The method belongs to the class of (nonlinear) algebraic multilevel iterations (AMLI). The crucial ingredient for obtaining robustness is the construction of a nested sequence of spaces based on local generalized eigenvalue problems. The method is analyzed in a rather general setting and is applied to the scalar elliptic equation, the equations of linear elasticity, and equations arising in the solution of Maxwell’s equations. Numerical results for the scalar elliptic equation are presented showing its robust convergence behavior and large coarsening factors in the sequence of nested spaces.
Application of a conservative, generalized multiscale finite element method to flow models
 J. Comput. Appl. Math
"... element method to flow models ..."
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A Multiscale Model Reduction Method for Partial Differential Equations
"... We propose a multiscale model reduction method for partial differential equations. The main purpose of this method is to derive an effective equation for multiscale problems without scale separation. An essential ingredient of our method is to decompose the harmonic coordinates into a smooth part an ..."
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We propose a multiscale model reduction method for partial differential equations. The main purpose of this method is to derive an effective equation for multiscale problems without scale separation. An essential ingredient of our method is to decompose the harmonic coordinates into a smooth part and a highly oscillatory part so that the smooth part is invertible and the highly oscillatory part is small. Such decomposition plays a key role in our construction of the effective equation. We show that the solution to the effective equation is smooth in H 2, and can be approximated by a regular coarse mesh. When the multiscale problem has scale separation and a periodic structure, our method recovers the traditional homogenized equation. Furthermore, we provide error analysis for our method and show that the solution to the effective equation is close to the original multiscale solution in the H 1 norm. Numerical results are presented to demonstrate the accuracy and robustness of the proposed method for several multiscale problems without scale separation, including a problem with a high contrast coefficient. 1
A global Jacobian method for mortar discretizations of nonlinear porous media flows
, 2013
"... We describe a nonoverlapping domain decomposition algorithm for nonlinear porous media flows discretized with the multiscale mortar mixed finite element method. There are two main ideas: (1) linearize the global system in both subdomain and interface variables simultaneously to yield a single Newto ..."
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We describe a nonoverlapping domain decomposition algorithm for nonlinear porous media flows discretized with the multiscale mortar mixed finite element method. There are two main ideas: (1) linearize the global system in both subdomain and interface variables simultaneously to yield a single Newton iteration; and (2) algebraically eliminate subdomain velocities (and optionally, subdomain pressures) to solve linear systems for the 1st (or the 2nd) Schur complements. Solving the 1st Schur complement system gives the multiscale solution without the need to solve an interface iteration. Solving the 2nd Schur complement system gives a linear interface problem for a nonlinear model. The methods are less complex than a previously developed nonlinear mortar algorithm, which requires two nested Newton iterations and a forward difference approximation. Furthermore, efficient linear preconditioners can be applied to speed up the iteration. The methods are implemented in parallel, and a numerical study is performed to demonstrate convergence behavior and parallel efficiency.
ROBUST MULTILEVEL SOLVERS FOR HIGHCONTRAST ANISOTROPIC MULTISCALE PROBLEMS
"... Abstract. A robust multilevel method for computing the solution of a scalar elliptic equation with anisotropic highly varying tensor coefficients is presented. The method, which belongs to the class of nonlinear algebraic multilevel iterations (AMLI), uses an abstract framework for general symmetric ..."
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Abstract. A robust multilevel method for computing the solution of a scalar elliptic equation with anisotropic highly varying tensor coefficients is presented. The method, which belongs to the class of nonlinear algebraic multilevel iterations (AMLI), uses an abstract framework for general symmetric positive definite bilinear forms previously presented in another publication by the author. The crucial ingredient for obtaining robustness with respect to the variations in the coefficients and the anisotropies is the design of a nested sequence of spaces based on local generalized eigenvalue problems. A discussion on how to achieve large coarsening factors in this sequence of spaces, which is desirable in terms of computational complexity, is included. Particular emphasis is put on how to handle the situation when the computed generating sets of the nested spaces are not minimal, i.e., do not constitute bases. Several numerical examples are provided verifying the theoretically established robustness results. 1.