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15
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...
Feti And NeumannNeumann Iterative Substructuring Methods: Connections And New Results
 Comm. Pure Appl. Math
, 1999
"... The FETI and NeumannNeumann families of algorithms are among the best known and most severely tested domain decomposition methods for elliptic partial differential equations. They are iterative substructuring methods and have many algorithmic components in common but there are also differences. The ..."
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Cited by 60 (17 self)
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The FETI and NeumannNeumann families of algorithms are among the best known and most severely tested domain decomposition methods for elliptic partial differential equations. They are iterative substructuring methods and have many algorithmic components in common but there are also differences. The purpose of this paper is to further unify the theory for these two families of methods and to introduce a new family of FETI algorithms. Bounds on the rate of convergence, which are uniform with respect to the coefficients of a family of elliptic problems with heterogeneous coefficients, are established for these new algorithms. The theory for a variant of the NeumannNeumann algorithm is also redeveloped stressing similarities to that for the FETI methods.
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.
Multilevel Schwarz Methods For Elliptic Problems With Discontinuous Coefficients In Three Dimensions
 NUMER. MATH
, 1994
"... Multilevel Schwarz methods are developed for a conforming finite element approximation of second order elliptic problems. We focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions. We establish a condition number estimate fo ..."
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Cited by 54 (13 self)
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Multilevel Schwarz methods are developed for a conforming finite element approximation of second order elliptic problems. We focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions. We establish a condition number estimate for the iterative operator, which is independent of the coefficients, and grows at most as the square of the number of levels. We also characterize a class of distributions of the coefficients, called quasimonotone, for which the weighted L²projection is stable and for which we can use the standard piecewise linear functions as a coarse space. In this case, we obtain optimal methods, i.e. bounds which are independent of the number of levels and subregions. We also design and analyze multilevel methods with new coarse spaces given by simple explicit formulas. We consider nonuniform meshes and conclude by an analysis of multilevel iterative substructuring methods.
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...
Intergrid Transfer Operators and Multilevel Preconditioners for Nonconforming Discretizations
, 1997
"... this paper discusses only multigrid (multiplicative) preconditioners and methods). The numbers for the Zienkiewicz element are much more encouraging which makes this element interesting for preconditioning finite element discretization matrices for fourth order problems. ..."
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Cited by 12 (4 self)
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this paper discusses only multigrid (multiplicative) preconditioners and methods). The numbers for the Zienkiewicz element are much more encouraging which makes this element interesting for preconditioning finite element discretization matrices for fourth order problems.
Adaptive Multilevel Iterative Techniques for Nonconforming Finite Element Discretizations
 EastWest J. Numer. Math
, 1995
"... We consider adaptive multilevel methods for the nonconforming P1 finite element approximation of linear second order elliptic boundary value problems. Emphasis is on the efficient solution of the discretized problems by multilevel preconditioned conjugate gradient iterations with respect to an adapt ..."
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Cited by 9 (6 self)
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We consider adaptive multilevel methods for the nonconforming P1 finite element approximation of linear second order elliptic boundary value problems. Emphasis is on the efficient solution of the discretized problems by multilevel preconditioned conjugate gradient iterations with respect to an adaptively generated hierarchy of possibly highly nonuniform triangulations. Local refinement of the elements of the triangulations is done by means of an efficient and reliable elementoriented a posteriori error estimator that can be derived by a defect correction in a higher order ansatz space and its hierarchical twolevel splitting. The performance of the preconditioners and the error estimator is illustrated by some test examples. Further, numerical results are given for the reverse biased pnjunction in semiconductor device simulation and the twogroup diffusion equations modeling the neutron fluxes in nuclear reactors. Keywords: nonconforming finite elements, multilevel preconditioners, ...
A Nonnested Coarse Space for Schwarz Type Domain Decomposition Methods
, 1993
"... In this paper, we develop a new technique, and a corresponding theory, for Schwarz type overlapping domain decomposition methods for solving large sparse linear systems which arise from finite element discretization of elliptic partial differential equations. The theory provides an optimal convergen ..."
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Cited by 6 (1 self)
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In this paper, we develop a new technique, and a corresponding theory, for Schwarz type overlapping domain decomposition methods for solving large sparse linear systems which arise from finite element discretization of elliptic partial differential equations. The theory provides an optimal convergence of an additive Schwarz algorithm that is constructed with a nonnested coarse space, and a not necessarily shape regular subdomain partitioning. It allows the use of subdomains with nonuniform aspect ratio and nonsmooth boundaries. The theory is also applicable to the overlapping graph partitioning algorithms recently developed by Cai and Saad [5], and to the nonnested coarse space method, such as these has been used by Cai, Gropp, Keyes and Tidriri [4] successfully for solving a nonlinear equation of aerodynamics. 1 Introduction Considerable interest has developed in Schwarz type overlapping domain decomposition methods for the numerical solution of partial differential equations, se...
Substructuring Preconditioning for Finite Element Approximations of Second Order Elliptic Problems. I. Nonconforming Linear Elements for the Poisson Equation in a Parallelepiped
 in parallelepiped, ISC Preprint #2, TAMU
, 1994
"... this paper, we use boldfaced letters to denote vectors in general in the space IR ..."
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Cited by 6 (2 self)
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this paper, we use boldfaced letters to denote vectors in general in the space IR