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Feti And NeumannNeumann Iterative Substructuring Methods: Connections And New Results
 Comm. Pure Appl. Math
, 1999
"... The FETI and NeumannNeumann families of algorithms are among the best known and most severely tested domain decomposition methods for elliptic partial differential equations. They are iterative substructuring methods and have many algorithmic components in common but there are also differences. The ..."
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Cited by 60 (17 self)
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The FETI and NeumannNeumann families of algorithms are among the best known and most severely tested domain decomposition methods for elliptic partial differential equations. They are iterative substructuring methods and have many algorithmic components in common but there are also differences. The purpose of this paper is to further unify the theory for these two families of methods and to introduce a new family of FETI algorithms. Bounds on the rate of convergence, which are uniform with respect to the coefficients of a family of elliptic problems with heterogeneous coefficients, are established for these new algorithms. The theory for a variant of the NeumannNeumann algorithm is also redeveloped stressing similarities to that for the FETI methods.
Graph Partitioning Algorithms With Applications To Scientific Computing
 Parallel Numerical Algorithms
, 1997
"... Identifying the parallelism in a problem by partitioning its data and tasks among the processors of a parallel computer is a fundamental issue in parallel computing. This problem can be modeled as a graph partitioning problem in which the vertices of a graph are divided into a specified number of su ..."
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Cited by 41 (0 self)
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Identifying the parallelism in a problem by partitioning its data and tasks among the processors of a parallel computer is a fundamental issue in parallel computing. This problem can be modeled as a graph partitioning problem in which the vertices of a graph are divided into a specified number of subsets such that few edges join two vertices in different subsets. Several new graph partitioning algorithms have been developed in the past few years, and we survey some of this activity. We describe the terminology associated with graph partitioning, the complexity of computing good separators, and graphs that have good separators. We then discuss early algorithms for graph partitioning, followed by three new algorithms based on geometric, algebraic, and multilevel ideas. The algebraic algorithm relies on an eigenvector of a Laplacian matrix associated with the graph to compute the partition. The algebraic algorithm is justified by formulating graph partitioning as a quadratic assignment p...
A nonoverlapping domain decomposition method for Maxwellâ€™s equations in three dimensions
 SIAM J. Numer. Anal
"... Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with muc ..."
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Cited by 35 (10 self)
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Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with much simpler coarse solvers. Though the condition number of the preconditioned system may not have a good bound, we are able to show that the convergence rate of the PCG method with such substructuring preconditioner is nearly optimal, and also robust with respect to the (possibly large) jumps of the coefficient in the elliptic equation. 1.
A NeumannNeumann Domain Decomposition Algorithm for Solving Plate and Shell Problems
 SIAM J. NUMER. ANAL
, 1997
"... We present a new NeumannNeumann type preconditioner of large scale linear systems arising from plate and shell problems. The advantage of the new method is a smaller coarse space than those of earlier method of the authors; this improves parallel scalability. A new abstract framework for NeumannNe ..."
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Cited by 33 (8 self)
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We present a new NeumannNeumann type preconditioner of large scale linear systems arising from plate and shell problems. The advantage of the new method is a smaller coarse space than those of earlier method of the authors; this improves parallel scalability. A new abstract framework for NeumannNeumann preconditioners is used to prove almost optimal convergence properties of the method. The convergence estimates are independent of the number of subdomains, coefficient jumps between subdomains, and depend only polylogarithmically on the number of elements per subdomain. We formulate and prove an approximate parametric variational principle for ReissnerMindlin elements as the plate thickness approaches zero, which makes the results applicable to a large class of nonlocking elements in everyday engineering use. The theoretical results are confirmed by computational experiments on model problems as well as examples from real world engineering practice.
Algebraic twolevel preconditioners for the Schur complement method
 SIAM J. SCIENTIFIC COMPUTING
, 1998
"... The solution of elliptic problems is challenging on parallel distributed memory computers as their Green's functions are global. To address this issue, we present a set of preconditioners for the Schur complement domain decomposition method. They implement a global coupling mechanism, through coarse ..."
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Cited by 16 (9 self)
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The solution of elliptic problems is challenging on parallel distributed memory computers as their Green's functions are global. To address this issue, we present a set of preconditioners for the Schur complement domain decomposition method. They implement a global coupling mechanism, through coarse space components, similar to the one proposed in [3]. The definition of the coarse space components is algebraic, they are defined using the mesh partitioning information and simple interpolation operators. These preconditioners are implemented on distributed memory computers without introducing any new global synchronization in the preconditioned conjugate gradient iteration. The numerical and parallel scalability of those preconditioners is illustrated on twodimensional model examples that have anisotropy and/or discontinuity phenomena.
Domain decomposition preconditioners for linearâ€“quadratic elliptic optimal control problems
, 2004
"... ABSTRACT. We develop and analyze a class of overlapping domain decomposition (DD) preconditioners for linearquadratic elliptic optimal control problems. Our preconditioners utilize the structure of the optimal control problems. Their execution requires the parallel solution of subdomain linearquad ..."
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Cited by 12 (4 self)
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ABSTRACT. We develop and analyze a class of overlapping domain decomposition (DD) preconditioners for linearquadratic elliptic optimal control problems. Our preconditioners utilize the structure of the optimal control problems. Their execution requires the parallel solution of subdomain linearquadratic elliptic optimal control problems, which are essentially smaller subdomain copies of the original problem. This work extends to optimal control problems the application and analysis of overlapping DD preconditioners, which have been used successfully for the solution of single PDEs. We prove that for a class of problems the performance of the twolevel versions of our preconditioners is independent of the mesh size and of the subdomain size. 1.
A preconditioner for the Schur complement domain decomposition method
 FOURTEENTH INTERNATIONAL CONFERENCE ON DOMAIN DECOMPOSITION METHODS
, 2003
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Local Preconditioners for TwoLevel NonOverlapping Domain Decomposition Methods
, 1999
"... We consider additive twolevel preconditioners, with a local and a global component, for the Schur complement system arising in nonoverlapping domain decomposition methods. We propose two new parallelizable local preconditioners. The rst one is a computationally cheap but numerically relevant alter ..."
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Cited by 9 (6 self)
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We consider additive twolevel preconditioners, with a local and a global component, for the Schur complement system arising in nonoverlapping domain decomposition methods. We propose two new parallelizable local preconditioners. The rst one is a computationally cheap but numerically relevant alternative to the classical block Jacobi preconditioner. The second one exploits all the information from the local Schur complement matrices and demonstrates an attractive numerical behavior on heterogeneous and anisotropic problems. We also propose two implementations based on approximate Schur complement matrices that are cheaper alternatives to construct the given preconditioners but that preserve their good numerical behavior. We compare their numerical performance with wellknown robust preconditioners such as BPS [6] and the balanced NeumannNeumann method [15].
Preconditioning the FETI Method for Problems with Intra and InterSubdomain Coefficient Jumps
 Ninth International Conference of Domain Decomposition Methods, 1997. URL = http://www.ddm.org/DD9
, 1997
"... this paper, we revisit both issues and present a preconditioning algorithm that addresses the problems of arbitrary subdomain aspect ratios, and large inter as well as intrasubdomain coefficient jumps (so far, most authors have addressed only the problem of intersubdomain coefficient jumps [LeT94 ..."
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Cited by 9 (0 self)
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this paper, we revisit both issues and present a preconditioning algorithm that addresses the problems of arbitrary subdomain aspect ratios, and large inter as well as intrasubdomain coefficient jumps (so far, most authors have addressed only the problem of intersubdomain coefficient jumps [LeT94]). The proposed preconditioner is derived from sound energy principles that were initially introduced in [RF96] for improving the accuracy of the solution of subdomain problems by polynomial and piecewise polynomial Lagrange multipliers. It can be equally used with the FETI and Balanced algorithms. However, because of space limitation, we limit our presentation to the case of the FETI method. We do not offer a mathematical proof of the optimality of our preconditioner, but we demonstrate numerically its scalability with the solution of highly heterogeneous structural mechanics problems. 2 The Focus Problem The solution of a problem of the form Ku = f , where K is a symmetric positive definite matrix arising from the discretization of some second or fourthorder elliptic Ninth International Conference on Domain Decomposition Methods Editor Petter E. Bjrstad, Magne S. Espedal and David E. Keyes c fl1998 DDM.org PRECONDITIONING FOR HETEROGENEOUS PROBLEMS 473 problem on a domain\Omega\Gamma can be obtained by partitioning\Omega into N s subdomains\Omega (s) , and gluing these with discrete Lagrange multipliers : K (s) u (s) = f (s) \Gamma B (s) T s = 1; :::; N s (2.1) s=Ns X s=1 B (s) u (s) = 0 (2.2) Here, B (s) is a signed subdomain Boolean matrix that extracts and signs the interface components of a vector or a matrix related to\Omega (s) . Eliminating u (s) from Eqs. (2.12.2) leads to the socalled dual interface problem F I \GammaG \Ga...
Balancing NeumannNeumann Preconditioners for the Mixed Formulation of AlmostIncompressible Linear Elasticity
, 2003
"... Balancing NeumannNeumann methods are extended to the equations arising from the mixed formulation of almostincompressible linear elasticity problems discretized with discontinuouspressure finite elements. This family of domain decomposition algorithms has previously been shown to be e#ective for ..."
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Cited by 8 (2 self)
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Balancing NeumannNeumann methods are extended to the equations arising from the mixed formulation of almostincompressible linear elasticity problems discretized with discontinuouspressure finite elements. This family of domain decomposition algorithms has previously been shown to be e#ective for large finite element approximations of positive definite elliptic problems. Our methods are proved to be scalable and to depend weakly on the size of the local problems. Our work is an extension of previous work by Pavarino and Widlund on BNN methods for Stokes equation.