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76
A restricted additive Schwarz preconditioner for general sparse linear systems
 SIAM J. Sci. Comput
, 1999
"... Abstract. We introduce some cheaper and faster variants of the classical additive Schwarz preconditioner (AS) for general sparse linear systems and show, by numerical examples, that the new methods are superior to AS in terms of both iteration counts and CPU time, as well as the communication cost w ..."
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Cited by 129 (24 self)
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Abstract. We introduce some cheaper and faster variants of the classical additive Schwarz preconditioner (AS) for general sparse linear systems and show, by numerical examples, that the new methods are superior to AS in terms of both iteration counts and CPU time, as well as the communication cost when implemented on distributed memory computers. This is especially true for harder problems such as indefinite complex linear systems and systems of convectiondiffusion equations from threedimensional compressible flows. Both sequential and parallel results are reported. Key words. Overlapping domain decomposition, preconditioner, iterative method, sparse matrix AMS(MOS) subject classifications. 65N30, 65F10
FETI and Neumann–Neumann iterative substructuring methods: connections and new results.
 Comm. Pure Appl. Math.,
, 2001
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Dualprimal FETI methods for threedimensional elliptic problems with heterogeneous coefficients
 SIAM J. Numer. Anal
, 2002
"... Abstract. In this paper, certain iterative substructuring methods with Lagrange multipliers are considered for elliptic problems in three dimensions. The algorithms belong to the family of dual{ primal FETI methods which have recently been introduced and analyzed successfully for elliptic problems i ..."
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Cited by 77 (14 self)
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Abstract. In this paper, certain iterative substructuring methods with Lagrange multipliers are considered for elliptic problems in three dimensions. The algorithms belong to the family of dual{ primal FETI methods which have recently been introduced and analyzed successfully for elliptic problems in the plane. The family of algorithms for three dimensions is extended and a full analysis is provided for the new algorithms. Particular attention is paid to nding algorithms with a small primal subspace since that subspace represents the only global part of the dual{primal preconditioner. It is shown that the condition numbers of several of the dual{primal FETI methods can be bounded polylogarithmically as a function of the dimension of the individual subregion problems and that the bounds are otherwise independent of the number of subdomains, the mesh size, and jumps in the coeÆcients. These results closely parallel those for other successful iterative substructuring methods of primal as well as dual type.
Overlapping nonmatching grids mortar element methods for elliptic problems
 SIAM J. Numer. Anal
, 1999
"... Abstract. In the first part of the paper, we introduce an overlapping mortar finite element methods for solving twodimensional elliptic problems discretized on overlapping nonmatching grids. We prove an optimal error bound and estimate the condition numbers of certain overlapping Schwarz precondit ..."
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Cited by 32 (7 self)
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Abstract. In the first part of the paper, we introduce an overlapping mortar finite element methods for solving twodimensional elliptic problems discretized on overlapping nonmatching grids. We prove an optimal error bound and estimate the condition numbers of certain overlapping Schwarz preconditioned systems for the twosubdomain case. We show that the error bound is independent of the size of the overlap and the ratio of the mesh parameters. In the second part, we introduce three additive Schwarz preconditioned conjugate gradient algorithms based on the trivial and harmonic extensions. We provide estimates for the spectral bounds on the condition numbers of the preconditioned operators. We show that although the error bound is independent of the size of the overlap, the condition number does depend on it. Numerical examples are presented to support our theory.
Adaptive Multilevel Methods for Edge Element Discretizations of Maxwell's Equations
, 1997
"... . The focus of this paper is on the efficient solution of boundary value problems involving the doublecurl operator. Those arise in the computation of electromagnetic fields in various settings, for instance when solving the electric or magnetic wave equation with implicit timestepping, when tackl ..."
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Cited by 32 (12 self)
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. The focus of this paper is on the efficient solution of boundary value problems involving the doublecurl operator. Those arise in the computation of electromagnetic fields in various settings, for instance when solving the electric or magnetic wave equation with implicit timestepping, when tackling timeharmonic problems or in the context of eddycurrent computations. Their discretization is based on on N'ed'elec's H(curl;\Omega\Gamma7131/59948 edge elements on unstructured grids. In order to capture local effects and to guarantee a prescribed accuracy of the approximate solution adaptive refinement of the grid controlled by a posteriori error estimators is employed. The hierarchy of meshes created through adaptive refinement forms the foundation for the fast iterative solution of the resulting linear systems by a multigrid method. The guiding principle underlying the design of both the error estimators and the multigrid method is the separate treatment of the kernel of the cu...
Iterative Substructuring Methods For Spectral Elements: Problems In Three Dimensions Based On Numerical Quadrature
 Courant Institute of Mathematical Sciences, Department of Computer Science
, 1994
"... . Iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems. Two preconditioners for pversion finite element methods based on continuous, piecewise Qp functions are considered for second order elliptic problems in three dimensi ..."
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Cited by 27 (9 self)
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. Iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems. Two preconditioners for pversion finite element methods based on continuous, piecewise Qp functions are considered for second order elliptic problems in three dimensions; these special methods can also be viewed as spectral element methods. The first iterative method is designed for the Galerkin formulation of the problem. The second applies to linear systems for a discrete model derived by using GaussLobattoLegendre quadrature. For both methods, it is established that the condition number of the relevant operator grows only in proportion to (1 + log p) 2 : These bounds are independent of the number of elements, into which the given region has been divided, their diameters, as well as the jumps in the coefficients of the elliptic equation between elements. Results of numerical computations are also given, which provide upper bounds on the condition n...
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.
Domain decomposition for less regular subdomains: Overlapping Schwarz in two dimensions
 SIAM J. Numer. Anal
"... Abstract. In the theory of domain decomposition methods, it is often assumed that each subdomain is the union of a small set of coarse triangles or tetrahedra. In this study, extensions to the existing theory which accommodate subdomains with much less regular shapes are presented; the subdomains ar ..."
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Cited by 22 (10 self)
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Abstract. In the theory of domain decomposition methods, it is often assumed that each subdomain is the union of a small set of coarse triangles or tetrahedra. In this study, extensions to the existing theory which accommodate subdomains with much less regular shapes are presented; the subdomains are required only to be John domains. Attention is focused on overlapping Schwarz preconditioners for problems in two dimensions with a coarse space component of the preconditioner, which allows for good results even for coefficients which vary considerably. It is shown that the condition number of the domain decomposition method is bounded by C(1 + H/δ)(1 + log(H/h)) 2, where the constant C is independent of the number of subdomains and possible jumps in coefficients between subdomains. Numerical examples are provided which confirm the theory and demonstrate very good performance of the method for a variety of subregions including those obtained when a mesh partitioner is used for the domain decomposition.
An optimal adaptive finite element method
, 2003
"... • model problem + (A)FEM • newest vertex bisection • convergence of AFEM ..."
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Cited by 19 (2 self)
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• model problem + (A)FEM • newest vertex bisection • convergence of AFEM
On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients
 SIAM J. Sci. Comput
, 2002
"... Abstract. The successful implementation of adaptive finite element methods based on a posteriori error estimates depends on several ingredients: an a posteriori error indicator, a refinement/coarsening strategy, and the choice of various parameters. The objective of the paper is to examine the influ ..."
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Cited by 19 (9 self)
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Abstract. The successful implementation of adaptive finite element methods based on a posteriori error estimates depends on several ingredients: an a posteriori error indicator, a refinement/coarsening strategy, and the choice of various parameters. The objective of the paper is to examine the influence of these factors on the performance of adaptive finite element methods for a model problem: the linear elliptic equation with strongly discontinuous coefficients. We derive a new a posteriori error estimator which depends locally on the oscillations of the coefficients around singular points. Extensive numerical experiments are reported to support our theoretical results and to show the competitive behaviors of the proposed adaptive algorithm. Key words. a posteriori error estimators, adaptive algorithm, performance