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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.
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...
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 57 (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.
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...
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.
Domain Decomposition Algorithms For Mixed Methods For Second Order Elliptic Problems
 Math. Comp
"... . In this paper domain decomposition algorithms for mixed finite element methods for linear and quasilinear second order elliptic problems in IR 2 and IR 3 are developed. A convergence theory for twolevel and multilevel Schwarz methods applied to the algorithms under consideration is given, and ..."
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Cited by 20 (12 self)
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. In this paper domain decomposition algorithms for mixed finite element methods for linear and quasilinear second order elliptic problems in IR 2 and IR 3 are developed. A convergence theory for twolevel and multilevel Schwarz methods applied to the algorithms under consideration is given, and its extension to other substructuring methods such as vertex space and balancing domain decomposition methods is considered. 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 finite element methods for the same differential problems. Numerical experiments are presented to illustrate the present techniques. 1. Introduction. This is the second paper of a sequence where we develop and analyze efficient iterative algorithms for solving the linear system arising from mixed finite element methods for linear and quasilinear second order elliptic proble...
A Parallel Implementation Of An Iterative Substructuring Algorithm For Problems In Three Dimensions
 SIAM J. Sci. Comput
, 1993
"... . Numerical results from a parallel implementation of an iterative substructuring algorithm are reported. The algorithm is for solving scalar, selfadjoint elliptic partial differential equations in three dimensions. Results are given for two variants of the algorithm. In the first variant, exact in ..."
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Cited by 18 (0 self)
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. Numerical results from a parallel implementation of an iterative substructuring algorithm are reported. The algorithm is for solving scalar, selfadjoint elliptic partial differential equations in three dimensions. Results are given for two variants of the algorithm. In the first variant, exact interior solvers are used; in the second, one multigrid Vcycle is used to approximately solve the interior problems. The results are compared to theoretical behavior of the algorithm reported in previous work. Key words. domain decomposition, finite elements, iterative substructuring, parallel computing. AMS(MOS) subject classifications. 65F10, 65N30 1. Introduction. Much work on domain decomposition algorithms has focused on the abstract analysis of the algorithms, with less discussion of implementation issues and few nontrivial numerical results. This paper focuses on the implementation, on a stateoftheart parallel machine, of several iterative substructuring algorithms for elliptic p...
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 18 (6 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...