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45
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 200 (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.
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 144 (32 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...
Balancing domain decomposition for problems with large jumps in coefficients
 Math. Comp
, 1996
"... Abstract. The Balancing Domain Decomposition algorithm uses in each iteration solution of local problems on the subdomains coupled with a coarse problem that is used to propagate the error globally and to guarantee that the possibly singular local problems are consistent. The abstract theory introdu ..."
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Cited by 88 (10 self)
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Abstract. The Balancing Domain Decomposition algorithm uses in each iteration solution of local problems on the subdomains coupled with a coarse problem that is used to propagate the error globally and to guarantee that the possibly singular local problems are consistent. The abstract theory introduced recently by the firstnamed author is used to develop condition number bounds for conforming linear elements in two and three dimensions. The bounds are independent of arbitrary coefficient jumps between subdomains and of the number of subdomains, and grow only as the squared logarithm of the mesh size h. Computational experiments for two and threedimensional problems confirm the theory. 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 57 (11 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...
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 50 (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.
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 48 (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.
Additive Schwarz Methods for Elliptic Finite Element Problems in Three Dimensions
 Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations
, 1991
"... . Many domain decomposition algorithms and certain multigrid methods can be described and analyzed as additive Schwarz methods. When designing and analyzing domain decomposition methods, we encounter special difficulties in the case of three dimensions and if the coefficients are discontinuous and v ..."
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Cited by 26 (7 self)
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. Many domain decomposition algorithms and certain multigrid methods can be described and analyzed as additive Schwarz methods. When designing and analyzing domain decomposition methods, we encounter special difficulties in the case of three dimensions and if the coefficients are discontinuous and vary over a large range. In this paper, we first introduce a general framework for Schwarz methods. Three classes of applications are then considered: certain wire basket based iterative substructuring methods, NeumannNeumann algorithms with low dimensional, global subspaces and a modified form of a multilevel algorithm introduced by Bramble, Pasciak and Xu. Department of Mathematics, Warsaw University, 2 Banach Street, 00913 Warsaw, Poland. This work was supported in part by the National Science Foundation under Grant NSFCCR8903003, and in part by Polish Scientific Grant #R.R.I. 10. y Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, N.Y. 10012. This work was s...
Domain decomposition algorithms for mixed methods for second order elliptic problems
 Math. Comp
, 1996
"... Abstract. In this talk domain decomposition algorithms for mixed nite element methods for linear secondorder elliptic problems in IR2 and IR 3 are discussed. A convergence theory for twolevel and multilevel Schwarz methods applied to the algorithms under consideration is given. It is shown that th ..."
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Cited by 21 (12 self)
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Abstract. In this talk domain decomposition algorithms for mixed nite element methods for linear secondorder elliptic problems in IR2 and IR 3 are discussed. A convergence theory for twolevel and multilevel Schwarz methods applied to the algorithms under consideration is given. It is shown that the condition number of these iterative methods is bounded uniformly from above in the same manner as in the theory of domain decomposition methods for conforming and nonconforming nite element methods for the same di erential problems. Numerical experiments are presented to illustrate the present techniques. 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 20 (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...