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105
Schwarz Analysis Of Iterative Substructuring Algorithms For Elliptic Problems In Three Dimensions
 SIAM J. Numer. Anal
, 1993
"... . Domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations of partial differential equations. The preconditioners are constructed from exact or approximate sol ..."
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Cited by 110 (26 self)
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. Domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations of partial differential equations. The preconditioners are constructed from exact or approximate solvers for the same partial differential equation restricted to a set of subregions into which the given region has been divided. In addition, the preconditioner is often augmented by a coarse, second level approximation, that provides additional, global exchange of information, and which can enhance the rate of convergence considerably. The iterative substructuring methods, based on decompositions of the region into nonoverlapping subregions, form one of the main families of such algorithms. Many domain decomposition algorithms can conveniently be described and analyzed as Schwarz methods. These algorithms are fully defined in terms of a set of subspaces and auxiliary bilinear forms. A gener...
Domain Decomposition Algorithms With Small Overlap
, 1994
"... Numerical experiments have shown that twolevel Schwarz methods often perform very well even if the overlap between neighboring subregions is quite small. This is true to an even greater extent for a related algorithm, due to Barry Smith, where a Schwarz algorithm is applied to the reduced linear ..."
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Cited by 82 (11 self)
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Numerical experiments have shown that twolevel Schwarz methods often perform very well even if the overlap between neighboring subregions is quite small. This is true to an even greater extent for a related algorithm, due to Barry Smith, where a Schwarz algorithm is applied to the reduced linear system of equations that remains after that the variables interior to the subregions have been eliminated. In this paper, a supporting theory is developed.
Schwarz Methods of NeumannNeumann Type for ThreeDimensional Elliptic Finite Element Problems
 Comm. Pure Appl. Math
, 1995
"... . Several domain decomposition methods of NeumannNeumann type are considered for solving the large linear systems of algebraic equations that arise from discretizations of elliptic problems by finite elements. We will only consider problems in three dimensions. Several new variants of the basic alg ..."
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Cited by 78 (17 self)
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. Several domain decomposition methods of NeumannNeumann type are considered for solving the large linear systems of algebraic equations that arise from discretizations of elliptic problems by finite elements. We will only consider problems in three dimensions. Several new variants of the basic algorithm are introduced in a Schwarz method framework that provides tools which have already proven very useful in the design and analysis of other domain decomposition and multilevel methods. The NeumannNeumann algorithms have several advantages over other domain decomposition methods. The subregions, which define the subproblems, only share the boundary degrees of freedom with their neighbors. The subregions can also be of quite arbitrary shape and many of the major components of the preconditioner can be constructed from subprograms available in standard finite element program libraries. However, in its original form, the algorithm lacks a mechanism for global transportation of informatio...
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
Overlapping Schwarz Methods On Unstructured Meshes Using NonMatching Coarse Grids
 Numer. Math
, 1996
"... . We consider two level overlapping Schwarz domain decomposition methods for solving the finite element problems that arise from discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Standard finite element interpolation from the coarse to the fine grid may ..."
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Cited by 49 (17 self)
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. We consider two level overlapping Schwarz domain decomposition methods for solving the finite element problems that arise from discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Standard finite element interpolation from the coarse to the fine grid may be used. Our theory requires no assumption on the substructures which constitute the whole domain, so each substructure can be of arbitrary shape and of different size. The global coarse mesh is allowed to be nonnested to the fine grid on which the discrete problem is to be solved and both the coarse meshes and the fine meshes need not be quasiuniform. In addition, the domains defined by the fine and coarse grid need not be identical. The one important constraint is that the closure of the coarse grid must cover any portion of the fine grid boundary for which Neumann boundary conditions are given. In this general setting, our algorithms have the same optimal convergence rate of the usual ...
Domain decomposition algorithms for indefinite elliptic problems
 SIAM J. Sci. Stat. Comput
, 1992
"... Iterative methods for linear systems of algebraic equations arising from the finite element discretization of nonsymmetric and indefinite elliptic problems are considered. Methods previously known to work well for positive definite, symmetric problems are extended to certain nonsymmetric problems, w ..."
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Cited by 48 (16 self)
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Iterative methods for linear systems of algebraic equations arising from the finite element discretization of nonsymmetric and indefinite elliptic problems are considered. Methods previously known to work well for positive definite, symmetric problems are extended to certain nonsymmetric problems, which can also have some eigenvalues in the left half plane. We first consider an additive Schwarz method applied to linear, second order, symmetric or nonsymmetric, indefinite elliptic boundary value problems in two and three dimensions. An alternative linear system, which has the same solution as the original problem, is derived and this system is then solved by using GMRES, an iterative method of conjugate gradient type. In each iteration step, a coarse mesh finite element problem and a number of local problems are solved on small, overlapping subregions into which the original region is subdivided. We show that the rate of convergence is independent of the number of degrees of freedom and the number of local problems if the coarse mesh is fine enough. The performance of the method in two dimensions is illustrated by results of several numerical experiments. We also consider two other iterative method for solving the same class of elliptic problems in two dimensions. Using an observation of Dryja and Widlund, we show that the rate of convergence of certain iterative substructuring methods deteriorates only quite slowly when the local problems increase in size. A similar result is established for Yserentant’s hierarchical basis method.
Additive Schwarz algorithms for parabolic convectiondiffusion equations
 Numer. Math
, 1991
"... In this paper, we consider the solution of linear systems of algebraic equations that arise from parabolic finite element problems. We introduce three additive Schwarz type domain decomposition methods for general, not necessarily selfadjoint, linear, second order, parabolic partial differential equ ..."
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Cited by 41 (6 self)
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In this paper, we consider the solution of linear systems of algebraic equations that arise from parabolic finite element problems. We introduce three additive Schwarz type domain decomposition methods for general, not necessarily selfadjoint, linear, second order, parabolic partial differential equations and also study the convergence rates of these algorithms. The resulting preconditioned linear system of equations is solved by the generalized minimal residual method. Numerical results are also reported. Key words Schwarz’s alternating method, domain decomposition, parabolic convectiondiffusion equation, finite elements. AMS(MOS) subject classifications. 65N30, 65F10 1
A subspace preconditioning algorithm for eigenvector/eigenvalue computation
 Adv. Comput. Math
, 1996
"... We consider the problem of computing a modest number of the smallest eigenvalues along with orthogonal bases for the corresponding eigenspaces of a symmetric positive definite operator A defined on a finite dimensional real Hilbert space V. In our applications, the dimension of V is large and the co ..."
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Cited by 39 (6 self)
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We consider the problem of computing a modest number of the smallest eigenvalues along with orthogonal bases for the corresponding eigenspaces of a symmetric positive definite operator A defined on a finite dimensional real Hilbert space V. In our applications, the dimension of V is large and the cost of inverting A is prohibitive. In this paper, we shall develop an effective parallelizable technique for computing these eigenvalues and eigenvectors utilizing subspace iteration and preconditioning for A. Estimates will be provided which show that the preconditioned method converges linearly when used with a uniform preconditioner under the assumption that the approximating subspace is close enough to the span of desired eigenvectors. 1. Introduction. In this paper, we shall be concerned with computing a modest number of the smallest eigenvalues and their corresponding eigenvectors of a large symmetric illconditioned system. More explicitly, let A be a symmetric and positive definite linear operator on a Ndimensional real vector space V with inner product (·, ·)
Globalized NewtonKrylovSchwarz algorithms and software for parallel implicit CFD
 Int. J. High Performance Computing Applications
, 1998
"... Key words. NewtonKrylovSchwarz algorithms, parallel CFD, implicit methods Abstract. Implicit solution methods are important in applications modeled by PDEs with disparate temporal and spatial scales. Because such applications require high resolution with reasonable turnaround, parallelization is e ..."
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Cited by 36 (14 self)
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Key words. NewtonKrylovSchwarz algorithms, parallel CFD, implicit methods Abstract. Implicit solution methods are important in applications modeled by PDEs with disparate temporal and spatial scales. Because such applications require high resolution with reasonable turnaround, parallelization is essential. The pseudotransient matrixfree NewtonKrylovSchwarz (ΨNKS) algorithmic framework is presented as a widely applicable answer. This article shows that, for the classical problem of threedimensional transonic Euler flow about an M6 wing, ΨNKS can simultaneously deliver • globalized, asymptotically rapid convergence through adaptive pseudotransient continuation and Newton’s method; • reasonable parallelizability for an implicit method through deferred synchronization and favorable communicationtocomputation scaling in the Krylov linear solver; and • high perprocessor performance through attention to distributed memory and cache locality, especially through the Schwarz preconditioner. Two discouraging features of ΨNKS methods are their sensitivity to the coding of the underlying PDE discretization and the large number of parameters that must be selected to govern convergence. We therefore distill several recommendations from our experience and from our reading of the literature on various algorithmic components of ΨNKS, and we describe a freely available, MPIbased portable parallel software implementation of the solver employed here. 1. Introduction. Disparate
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.