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FETI and Neumann–Neumann iterative substructuring methods: Connections and new results
 Comm. on Pure Appl. Math
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A nonoverlapping domain decomposition method for Maxwell’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 46 (16 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 46 (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.
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 46 (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...
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 c ..."
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Cited by 19 (11 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 linear–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 19 (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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Prepartitioning as a Way to Mesh Subdomains in Parallel
 IN 5TH INTERNATIONAL MESHING ROUNDTABLE
, 1996
"... We introduce a new method to mesh a physical domain in parallel. Given a domain defined by a set of boundary facets, we construct a separator of it. As a result, we obtain several subdomains, each of them being defined by a set of boundary facets. Naturally, the separation step can be applied recu ..."
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Cited by 14 (2 self)
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We introduce a new method to mesh a physical domain in parallel. Given a domain defined by a set of boundary facets, we construct a separator of it. As a result, we obtain several subdomains, each of them being defined by a set of boundary facets. Naturally, the separation step can be applied recursively to increase the number of subdomains, or to reduce the size of one subdomain. Then, a classical mesh generator is applied to achieve the computation. Along with the theoretical background that allows to mesh such a separator (this process is called prepartitioning) , we show how the method performs on a large set of industrial examples. Some parallel simulations have been done, and the results for large examples are promising.
A MicroMacro And Parallel Computational Strategy For HighlyHeterogeneous Structures
, 2000
"... this paper. For a perfect interface, contains the transmission conditions for forces: Y Q Q@Y Q and for displacements: ('^ QXQZY V o V Q !" $ s jBj Y 5 + (U^ QXQZY ! V Q o Q@Y !" + s jBj Y also contains the boundary conditions for boundary interfaces inc ..."
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Cited by 13 (2 self)
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this paper. For a perfect interface, contains the transmission conditions for forces: Y Q Q@Y Q and for displacements: ('^ QXQZY V o V Q !" $ s jBj Y 5 + (U^ QXQZY ! V Q o Q@Y !" + s jBj Y also contains the boundary conditions for boundary interfaces included in or
Balancing NeumannNeumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity
 Numer. Math
"... Abstract. Balancing NeumannNeumann methods are extented to mixed formulations of the linear elasticity system with discontinuous coeÆcients, discretized with mixed nite or spectral elements with discontinuous pressures. These domain decomposition methods implicitly eliminate the degrees of freedom ..."
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Cited by 13 (3 self)
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Abstract. Balancing NeumannNeumann methods are extented to mixed formulations of the linear elasticity system with discontinuous coeÆcients, discretized with mixed nite or spectral elements with discontinuous pressures. These domain decomposition methods implicitly eliminate the degrees of freedom associated with the interior of each subdomain and solve iteratively the resulting saddle point Schur complement using a hybrid preconditioner based on a coarse mixed elasticity problem and local mixed elasticity problems with natural and essential boundary conditions. A polylogarithmic bound in the local number of degrees of freedom is proven for the condition number of the preconditioned operator in the constant coeÆcient case. Parallel and serial numerical experiments conrm the theoretical results, indicate that they still hold for systems with discontinuous coeÆcients, and show that our algorithm is scalable, parallel, and robust with respect to material heterogeneities. The results on heterogeneous general problems are also supported in part by our theory.