Results 1  10
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41
Balancing Domain Decomposition
 Comm. Numer. Meth. Engrg
, 1993
"... The NeumannNeumann algorithm is known to be an efficient domain decomposition preconditioner with unstructured subdomains for iterative solution of finite element discretizations of difficult problems with strongly discontinuous coefficients [6]. However, this algorithm suffers from the need to sol ..."
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Cited by 131 (11 self)
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The NeumannNeumann algorithm is known to be an efficient domain decomposition preconditioner with unstructured subdomains for iterative solution of finite element discretizations of difficult problems with strongly discontinuous coefficients [6]. However, this algorithm suffers from the need to solve in each iteration an inconsistent singular problem for every subdomain, and its convergence deteriorates with increasing number of subdomains due to the lack of a coarse problem to propagate the error globally. We show that the equilibrium conditions for the singular problems on subdomains lead to a simple and natural construction of a coarse problem. The construction is purely algebraic and applies also to systems, such as those that arize in elasticity. A convergence bound independent on the number of subdomains is proved and results of computational tests are reported.
Convergence of Algebraic Multigrid Based on Smoothed Aggregation
 Computing
, 1998
"... . We prove a convergence estimate for the Algebraic Multigrid Method with prolongations defined by aggregation using zero energy modes, followed by a smoothing. The method input is the problem matrix and a matrix of the zero energy modes. The estimate depends only polylogarithmically on the mesh siz ..."
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Cited by 90 (11 self)
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. We prove a convergence estimate for the Algebraic Multigrid Method with prolongations defined by aggregation using zero energy modes, followed by a smoothing. The method input is the problem matrix and a matrix of the zero energy modes. The estimate depends only polylogarithmically on the mesh size, and requires only a weak approximation property for the aggregates, which can be apriori verified computationally. Construction of the prolongator in the case of a general second order system is described, and the assumptions of the theorem are verified for a scalar problem discretized by linear conforming finite elements. Key words. Algebraic multigrid, zero energy modes, convergence theory, computational mechanics, Finite Elements, iterative solvers 1. Introduction. This paper is concerned with the analysis of an Algebraic Multigrid Method (AMG) based on smoothed aggregation, which we have introduced in [28], and which in turn is a further development of [25, 26]. This method and its ...
An Overlapping Schwarz Method for Spectral Element Solution of the Incompressible NavierStokes Equations
 J. Comp. Phys
, 1997
"... Efficient solution of the NavierStokes equations in complex domains is dependent upon the availability of fast solvers for sparse linear systems. For unsteady incompressible flows, the pressure operator is the leading contributor to stiffness, as the characteristic propagation speed is infinite. ..."
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Cited by 54 (23 self)
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Efficient solution of the NavierStokes equations in complex domains is dependent upon the availability of fast solvers for sparse linear systems. For unsteady incompressible flows, the pressure operator is the leading contributor to stiffness, as the characteristic propagation speed is infinite. In the context of operator splitting formulations, it is the pressure solve which is the most computationally challenging, despite its elliptic origins. We examine several preconditioners for the consistent L 2 Poisson operator arising in the lP N \Gamma lP N \Gamma2 spectral element formulation of the incompressible NavierStokes equations. We develop a finite element based additive Schwarz preconditioner using overlapping subdomains plus a coarse grid projection operator which is applied directly to the pressure on the interior Gauss points. For large twodimensional problems this approach can yield as much as a fivefold reduction in simulation time over previously employed methods based upon deflation. To appear in J. of Comp. Phys., 1997. Present address: Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. Email: pff@cfm.brown.edu 1 1
On the convergence of a dualprimal substructuring method
, 2000
"... Abstract. In the DualPrimal FETI method, introduced by Farhat et al. [5], the domain is decomposed into nonoverlapping subdomains, but the degrees of freedom on crosspoints remain common to all subdomains adjacent to the crosspoint. The continuity of the remaining degrees of freedom on subdomain i ..."
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Cited by 38 (4 self)
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Abstract. In the DualPrimal FETI method, introduced by Farhat et al. [5], the domain is decomposed into nonoverlapping subdomains, but the degrees of freedom on crosspoints remain common to all subdomains adjacent to the crosspoint. The continuity of the remaining degrees of freedom on subdomain interfaces is enforced by Lagrange multipliers and all degrees of freedom are eliminated. The resulting dual problem is solved by preconditioned conjugate gradients. We give an algebraic bound on the condition number, assuming only a single inequality in discrete norms, and use the algebraic bound to show that the condition number is bounded by C(1 + log2 (H/h)) for both second and fourth order elliptic selfadjoint problems discretized by conforming finite elements, as well as for a wide class of finite elements for the ReissnerMindlin plate model. 1. Introduction. This
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 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.
Some Nonoverlapping Domain Decomposition Methods
, 1998
"... . The purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving secondorder elliptic problems in two and three dimensions. The methods under scrutiny fall into two major categories: the substructuringtype methods and the Neumann ..."
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Cited by 35 (6 self)
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. The purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving secondorder elliptic problems in two and three dimensions. The methods under scrutiny fall into two major categories: the substructuringtype methods and the NeumannNeumanntype methods. The basic framework used for analysis is the parallel subspace correction method or additive Schwarz method, and other technical tools include localglobal and globallocal techniques. The analyses for both two and threedimensional cases are carried out simultaneously. Some internal relationships between various algorithms are observed and several new variants of the algorithms are also derived. Key words. nonoverlapping domain decomposition, Schur complement, localglobal and globallocal techniques, jumps in coe#cients, substructuring, NeumannNeumann, balancing methods AMS subject classifications. 65N30, 65N55, 65F10 PII. S0036144596306800 1. Introduction. T...
Terascale spectral element algorithms and implementations, in Supercomputing ’99
 Proceedings of the 1999 ACM/IEEE conference on Supercomputing (CDROM
, 1999
"... We describe the development and implementation of an efficient spectral element code for multimillion gridpoint simulations of incompressible flows in general two and threedimensional domains. Key to this effort has been the development of scalable solvers for elliptic problems and a stabilization ..."
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Cited by 22 (10 self)
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We describe the development and implementation of an efficient spectral element code for multimillion gridpoint simulations of incompressible flows in general two and threedimensional domains. Key to this effort has been the development of scalable solvers for elliptic problems and a stabilization scheme that admits full use of the method’s highorder accuracy. We review these and other recently developed algorithmic underpinnings that have resulted in good parallel and vector performance on a broad range of architectures and that, with sustained performance of 319 GFLOPS on 2048 nodes of the Intel ASCIRed machine at Sandia, readies us for the multithousand node terascale computing systems now coming on line at the DOE labs. 1
Analysis Of Lagrange Multiplier Based Domain Decomposition
, 1998
"... The convergence of a substructuring iterative method with Lagrange multipliers known as Finite Element Tearing and Interconnecting (FETI) method is analyzed in this thesis. This method, originally proposed by Farhat and Roux, decomposes finite element discretization of an elliptic boundary value pro ..."
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Cited by 17 (5 self)
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The convergence of a substructuring iterative method with Lagrange multipliers known as Finite Element Tearing and Interconnecting (FETI) method is analyzed in this thesis. This method, originally proposed by Farhat and Roux, decomposes finite element discretization of an elliptic boundary value problem into Neumann problems on the subdomains, plus a coarse problem for the subdomain null space components. For linear conforming elements and preconditioning by Dirichlet problems on the subdomains, the asymptotic bound on the condition number C(1 log(H=h)) fl , where fl = 2 or 3, is proved for a second order problem, h denoting the characteristic element size and H the size of subdomains. A similar method proposed by Park is shown to be equivalent to FETI with a special choice of some components and the bound C(1 log(H=h)) 2 on the condition number is established. Next, the original FETI method is generalized to fourth order plate bending problems. The main idea there is to enfor...
A DualPrimal Feti Method For Incompressible Stokes Equations
, 2001
"... . In this paper, a dualprimal FETI method is developed for incompressible Stokes equations approximated by mixed nite elements with discontinuous pressures. The domain of the problem is decomposed into nonoverlapping subdomains, and the continuity of the velocity across the subdomain interface is e ..."
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Cited by 13 (2 self)
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. In this paper, a dualprimal FETI method is developed for incompressible Stokes equations approximated by mixed nite elements with discontinuous pressures. The domain of the problem is decomposed into nonoverlapping subdomains, and the continuity of the velocity across the subdomain interface is enforced by introducing Lagrange multipliers. By a Schur complement procedure, solving the indenite Stokes problem is reduced to solving a symmetric positive denite problem for the dual variables, i.e., the Lagrange multipliers. This dual problem is solved by a Krylov space method with a Dirichlet preconditioner. At each step of the iteration, both subdomain problems and a coarse problem on the coarse subdomain mesh are solved by a direct method. It is proved that the condition number of this preconditioned dual problem is independent of the number of subdomains and bounded from above by the product of the inverse of the infsup constant of the discrete problem and the square of the logarithm of the number of unknowns in the individual subdomain problems. Illustrative numerical results are presented by solving a lid driven cavity problem. Key words. domain decomposition, Stokes, FETI, dualprimal methods AMS subject classications. 65N30, 65N55, 76D07 1.
Partitioning for complex objectives
 In Proceedings of International Parallel and Distributed Processing Symposium
, 2001
"... Graph partitioning is an important tool for dividing work amongst processors of a parallel machine, but it is unsuitable for some important applications. Specifically, graph partitioning requires the work per processor to be a simple sum of vertex weights. For many applications, this assumption is n ..."
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Cited by 13 (4 self)
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Graph partitioning is an important tool for dividing work amongst processors of a parallel machine, but it is unsuitable for some important applications. Specifically, graph partitioning requires the work per processor to be a simple sum of vertex weights. For many applications, this assumption is not true — the work (or memory) is a complex function of the partition. In this paper we describe a general framework for addressing such partitioning problems and investigate its utility on two applications — partitioning so that overlapped subdomains are balanced and partitioning to minimize the sum of computation plus communication time. 1