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78
How Good is Recursive Bisection?
 SIAM J. Sci. Comput
, 1995
"... . The most commonly used pway partitioning method is recursive bisection (RB). It first divides a graph or a mesh into two equal sized pieces, by a "good" bisection algorithm, and then recursively divides the two pieces. Ideally, we would like to use an optimal bisection algorithm. Becaus ..."
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Cited by 86 (4 self)
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. The most commonly used pway partitioning method is recursive bisection (RB). It first divides a graph or a mesh into two equal sized pieces, by a "good" bisection algorithm, and then recursively divides the two pieces. Ideally, we would like to use an optimal bisection algorithm. Because the optimal bisection problem, that partitions a graph into two equal sized subgraphs to minimize the number of edges cut, is NPcomplete, practical RB algorithms use more efficient heuristics in place of an optimal bisection algorithm. Most such heuristics are designed to find the best possible bisection within allowed time. We show that the recursive bisection method, even when an optimal bisection algorithm is assumed, may produce a pway partition that is very far way from the optimal one. Our negative result is complemented by two positive ones: First we show that for some important classes of graphs that occur in practical applications, such as wellshaped finite element and finite difference...
FETI and Neumann{Neumann Iterative Substructuring Methods
 Connections and New Results. Comm. Pure Appl. Math
, 2001
"... ..."
A Mortar Finite Element Method Using Dual Spaces For The Lagrange Multiplier
 SIAM J. Numer. Anal
, 1998
"... The mortar finite element method allows the coupling of different discretization schemes and triangulations across subregion boundaries. In the original mortar approach the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modified trace space which ..."
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Cited by 56 (8 self)
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The mortar finite element method allows the coupling of different discretization schemes and triangulations across subregion boundaries. In the original mortar approach the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modified trace space which serves as a space of Lagrange multipliers. In this paper, this Lagrange multiplier space is replaced by a dual space without losing the optimality of the method. The advantage of this new approach is that the matching condition is much easier to realize. In particular, all the basis functions of the new method are supported in a few elements. The mortar map can be represented by a diagonal matrix; in the standard mortar method a linear system of equations must be solved. The problem is considered in a positive definite nonconforming variational as well as an equivalent saddlepoint formulation.
Optimized Schwarz methods
 SIAM Journal on Numerical Analysis
, 2006
"... Abstract. Optimized Schwarz methods are a new class of Schwarz methods with greatly enhanced convergence properties. They converge uniformly faster than classical Schwarz methods and their convergence rates dare asymptotically much better than the convergence rates of classical Schwarz methods if th ..."
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Cited by 48 (11 self)
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Abstract. Optimized Schwarz methods are a new class of Schwarz methods with greatly enhanced convergence properties. They converge uniformly faster than classical Schwarz methods and their convergence rates dare asymptotically much better than the convergence rates of classical Schwarz methods if the overlap is of the order of the mesh parameter, which is often the case in practical applications. They achieve this performance by using new transmission conditions between subdomains which greatly enhance the information exchange between subdomains and are motivated by the physics of the underlying problem. We analyze in this paper these new methods for symmetric positive definite problems and show their relation to other modern domain decomposition methods like the new Finite Element Tearing and Interconnect (FETI) variants.
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
Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity
, 1990
"... The use of the finite element method for elasticity problems results in extremely large, sparse linear systems. Historically these have been solved using direct solvers like Choleski's method. These linear systems are often illconditioned and hence require good preconditioners if they are to b ..."
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Cited by 40 (1 self)
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The use of the finite element method for elasticity problems results in extremely large, sparse linear systems. Historically these have been solved using direct solvers like Choleski's method. These linear systems are often illconditioned and hence require good preconditioners if they are to be solved iteratively. We propose and analyze three new, parallel iterative domain decomposition algorithms for the solution of these linear systems. The algorithms are also useful for other elliptic partial differential equations. Domain decomposition algorithms are designed to take advantage of a new generation of parallel computers. The domain is decomposed into overlapping or nonoverlapping subdomains. The discrete approximation to a partial differential equation is then obtained iteratively by solving problems associated with each subdomain. The algorithms are often accelerated using the conjugate gradient method. The first new algorithm presented here borrows heavily from multilevel type a...
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 36 (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 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 36 (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.
A Hierarchical Preconditioner For The Mortar Finite Element Method
 ETNA
, 1996
"... . Mortar elements form a family of nonconforming finite element methods that are more flexible than conforming finite elements and are known to be as accurate as their conforming counterparts. A fast iterative method is developed for linear, second order elliptic equations in the plane. Our algorith ..."
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Cited by 25 (1 self)
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. Mortar elements form a family of nonconforming finite element methods that are more flexible than conforming finite elements and are known to be as accurate as their conforming counterparts. A fast iterative method is developed for linear, second order elliptic equations in the plane. Our algorithm is modeled on a hierarchical basis preconditioner previously analyzed and tested, for the conforming case, by Barry Smith and the second author. A complete analysis and results of numerical experiments are given for lower order mortar elements and geometrically conforming decompositions of the region into subregions. Key words. domain decomposition, mortar finite element method, hierarchical preconditioner AMS(MOS) subject classifications. 65F30, 65N22, 65N30, 65N55 1. Introduction. Mortar finite element methods were introduced by Bernardi, Maday, and Patera; see [8]. The discretization of an elliptic, second order problem starts by partitioning the computational domain\Omega into the...
Stability Estimates of the Mortar Finite Element Method for 3Dimensional Problems
 EastWest J. Numer. Math
, 1998
"... This paper is concerned with the mortar finite element method for three spatial variables. The two main issues are the proof of the LBB condition based on appropriate choices of Lagrange multipliers and optimal efficiency of corresponding multigrid schemes for the whole coupled systems of equations. ..."
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Cited by 25 (3 self)
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This paper is concerned with the mortar finite element method for three spatial variables. The two main issues are the proof of the LBB condition based on appropriate choices of Lagrange multipliers and optimal efficiency of corresponding multigrid schemes for the whole coupled systems of equations. The implementation of the smoothing procedure also differs from that one used in the 2dimensional case. Key words: Mortar method, domain decomposition, saddle point problems, L 2  stability of mortar projections, multigrid algorithms, error estimates, efficiency of smoothing procedures. AMS subject classification: 65N55, 65N30, 65F10, 46E35. 1 Introduction The mortar method is a domain decomposition method with nonoverlapping subdomains, see e.g. [1, 2, 3, 6]. The matching of discretizations on adjacent subdomains is only enforced weakly which, in particular, facilitates employing different types of discretizations on different subdomains. Even in the case when only finite elements are ...