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Domain decomposition preconditioners for multiscale flows in high contrast media: reduced dimension coarse spaces. Multiscale Model
 Simul
"... We present a new class of coarse spaces for twolevel additive Schwarz preconditioners that yield condition number bound independent of the contrast in the media properties. These coarse spaces are an extension of the spaces discussed in Introduction Many problems in applied sciences occur in medi ..."
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Cited by 44 (15 self)
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We present a new class of coarse spaces for twolevel additive Schwarz preconditioners that yield condition number bound independent of the contrast in the media properties. These coarse spaces are an extension of the spaces discussed in Introduction Many problems in applied sciences occur in media that contains multiple scales and has high contrast in the properties. For example, it is very common to have several orders of magnitude of variations in the permeability field in natural porous formations. Domain decomposition preconditioners are often used to solve the finescale system that arises from the discretization of partial differential equations. The number of iterations required by domain decomposition preconditioners is typically affected by the contrast in the media properties that are within each coarsegrid block (e.g., The design and analysis of preconditioners that converge independent of the contrast is important for many applications, such as porous media flows where flow problems are solved multiple times. In
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
A NEW MULTISCALE FINITE ELEMENT METHOD FOR HIGHCONTRAST ELLIPTIC INTERFACE PROBLEMS
"... Abstract. We introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes, and without resolving the interfaces. A typical application would be the modelling of f ..."
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Cited by 26 (1 self)
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Abstract. We introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes, and without resolving the interfaces. A typical application would be the modelling of flow in a porous medium containing a number of inclusions of low (or high) permeability embedded in a matrix of high (respectively low) permeability. Our method is H 1 conforming, with degrees of freedom at the nodes of a triangular mesh and requires the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface, but uses standard linear approximation otherwise. A key point is the introduction of novel coefficientdependent boundary conditions for the subgrid problems. Under moderate assumptions, we prove that our methods have (optimal) convergence rate of O(h) in the energy norm and O(h 2) in the L2 norm where h is the (coarse) mesh diameter and the hidden constants in these estimates are independent of the “contrast ” (i.e. ratio of largest to smallest value) of the PDE coefficient. For standard elements the best estimate in the energy norm would be O(h 1/2−ε) with a hidden constant which in general depends on the contrast. The new interior boundary conditions depend not only on the contrast of the coefficients, but also on the angles of intersection of the interface with the element edges. 1.
Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms. Arxiv preprint arXiv:1105.1131
, 2011
"... Abstract. An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into “local ” subspaces and a global “coarse ” space is developed. Particular applications of this abstract framework include practically important probl ..."
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Cited by 26 (11 self)
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Abstract. An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into “local ” subspaces and a global “coarse ” space is developed. Particular applications of this abstract framework include practically important problems in porous media applications such as: the scalar elliptic (pressure) equation and the stream function formulation of its mixed form, Stokes’ and Brinkman’s equations. The constant in the corresponding abstract energy estimate is shown to be robust with respect to mesh parameters as well as the contrast, which is defined as the ratio of high and low values of the conductivity (or permeability). The derived stable decomposition allows to construct additive overlapping Schwarz iterative methods with condition numbers uniformly bounded with respect to the contrast and mesh parameters. The coarse spaces are obtained by patching together the eigenfunctions corresponding to the smallest eigenvalues of certain local problems. A detailed analysis of the abstract setting is provided. The proposed decomposition builds on a method of Efendiev and Galvis [Multiscale Model. Simul., 8 (2010), pp. 1461–1483] developed for second order scalar elliptic problems with high contrast. Applications to the finite element discretizations of the second order elliptic problem in Galerkin and mixed formulation, the Stokes equations, and Brinkman’s problem are presented. A number of numerical experiments for these problems in two spatial dimensions are provided. 1.
Coupled finite and boundary element tearing and interconnecting solvers for nonlinear potential problems
 Z. Angew. Math. Mech
"... Abstract. Finite element tearing and interconnecting (FETI) methods and boundary element tearing and interconnecting (BETI) methods are special iterative substructuring methods with Lagrange multipliers. For elliptic boundary value problems on bounded domains, the condition number of these method ..."
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Cited by 16 (10 self)
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Abstract. Finite element tearing and interconnecting (FETI) methods and boundary element tearing and interconnecting (BETI) methods are special iterative substructuring methods with Lagrange multipliers. For elliptic boundary value problems on bounded domains, the condition number of these methods can be rigorously bounded by C ((1+log(H/h))2, where H is the subdomain diameter and h the mesh size. The constant C is independent of H, h and possible jumps in the coefficients of the partial differential equation. In certain situations, e. g., in electromagnetic field computations, instead of imposing artificial boundary conditions one may be interested in modelling the real physical behaviour in an exterior domain with a radiation condition. In this work we analyze onelevel BETI methods for such unbounded domains and show explicit condition number estimates similar to the one above. Our theoretical results are confirmed in numerical experiments. 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
R.: Towards a rigorously justified algebraic preconditioner for highcontrast diffusion problems
 Comput. Vis. Sci
, 2008
"... Abstract. In this paper we present a new preconditioner suitable for solving linear systems arising from finite element approximations of elliptic PDEs with highcontrast coefficients. The construction of the preconditioner consists of two phases. The first phase is an algebraic one which partitions ..."
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Cited by 14 (8 self)
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Abstract. In this paper we present a new preconditioner suitable for solving linear systems arising from finite element approximations of elliptic PDEs with highcontrast coefficients. The construction of the preconditioner consists of two phases. The first phase is an algebraic one which partitions the freedoms into “high ” and “low ” permeability regions which may be of arbitrary geometry. This partition yields a corresponding blocking of the stiffness matrix and hence a formula for the action of its inverse involving the inverses of both the high permeability block and its Schur complement in the original matrix. The structure of the required subblock inverses in the high contrast case is revealed by a singular perturbation analysis (with the contrast playing the role of a large parameter). This shows that for high enough contrast each of the subblock inverses can be approximated well by solving only systems with constant coefficients. The second phase of the algorithm involves the approximation of these constant coefficient systems using multigrid methods. The result is a general method of algebraic character which (under suitable hypotheses) can be proved to be robust with respect to both the contrast and the mesh size. While a similar performance is also achieved in practice by algebraic multigrid (AMG) methods, this performance is still without theoretical justification. Since the first phase of our
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.
A family of energy minimizing coarse spaces for overlapping Schwarz preconditioners
 in Proceedings of the 17th International Conference on Domain Decomposition Methods in Science and Engineering
, 2007
"... Summary. A simple and effective approach is presented to construct coarse spaces for overlapping Schwarz preconditioners. The approach is based on energy minimizing extensions of coarse trace spaces, and can be viewed as a generalization of earlier work by Dryja, Smith, and Widlund. The use of thes ..."
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Cited by 11 (7 self)
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Summary. A simple and effective approach is presented to construct coarse spaces for overlapping Schwarz preconditioners. The approach is based on energy minimizing extensions of coarse trace spaces, and can be viewed as a generalization of earlier work by Dryja, Smith, and Widlund. The use of these coarse spaces in overlapping Schwarz preconditioners leads to condition numbers bounded by C(1 +H/δ)(1 + log(H/h)) for certain problems when coefficient jumps are aligned with subdomain boundaries. For problems without coefficient jumps, it is possible to remove the log(H/h) factor in this bound by a suitable enrichment of the coarse space. Comparisons are made with the coarse spaces of two other substructuring preconditioners. Numerical examples are also presented for a variety of problems.