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• 1
Electronic Resource
Springer
Numerische Mathematik 73 (1996), S. 473-487
ISSN: 0945-3245
Keywords: Mathematics Subject Classification (1991):65N30
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Summary. We analyze the convergence of a substructuring iterative method with Lagrange multipliers, proposed recently by Farhat and Roux. The method decomposes finite element discretization of an elliptic boundary value problem into Neumann problems on the subdomains plus a coarse problem for the subdomain nullspace components. For linear conforming elements and preconditioning by the Dirichlet problems on the subdomains, we prove the asymptotic bound on the condition number $C (1+\log (H/h))^\gamma$ , $\gamma=2$ or $3$ ,where $h$ is the characteristic element size and $H$ subdomain size.
Type of Medium: Electronic Resource
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• 2
Electronic Resource
Springer
Numerische Mathematik 46 (1985), S. 147-157
ISSN: 0945-3245
Keywords: AMS(MOS): 65J05 ; 65R20 ; CR: 61.9
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Summary We describe a unifying framework for multigrid methods and projection-iterative methods for integral equations of the second kind, and for the iterative aggregation method for solving input-output relations. The methods are formulated as iterations combined with a defect correction in a subspace. Convergence proofs use contraction arguments and thus involve the nonlinear case automatically. Some new results are presented.
Type of Medium: Electronic Resource
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• 3
Electronic Resource
Springer
Numerische Mathematik 58 (1990), S. 79-93
ISSN: 0945-3245
Keywords: AMS(MOS) ; 65N20 ; CR: G1.8
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Summary We study symmetric positive definite linear systems, with a 2-by-2 block matrix preconditioned by inverting directly one of the diagonal blocks and suitably preconditioning the other. Using an approximate version of Young's “Property A”, we show that the condition number of the Schur complement is smaller than the condition number obtained by the block-diagonal preconditioning. We also get bounds on both condition numbers from a strengthened Cauchy inequality. For systems arising from the finite element method, the bounds do not depend on the number of elements and can be obtained from element-by-element computations. The results are applied to thep-version finite element method, where the first block of variables consists of degrees of freedom of a low order.
Type of Medium: Electronic Resource
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• 4
Electronic Resource
Springer
Applied mathematics & optimization 11 (1984), S. 77-95
ISSN: 1432-0606
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract A fast iterative method for the solution of large, sparse, symmetric, positive definite linear complementarity problems is presented. The iterations reduce to linear iterations in a neighborhood of the solution if the problem is nondegenerate. The variational setting of the method guarantees global convergence. As an application, we consider a discretization of a Dirichlet obstacle problem by triangular linear finite elements. In contrast to usual iterative methods, the observed rate of convergence does not deteriorate with step size.
Type of Medium: Electronic Resource
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• 5
Electronic Resource
Springer
BIT 31 (1991), S. 76-88
ISSN: 1572-9125
Keywords: 65N30 ; 65J10 ; 35J20 ; 15A18 ; Orthogonal Projections ; Parallel Computing ; Domain Decomposition ; Grid Refinement ; Schwarz Alternating Method
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract Many parallel iterative algorithms for solving symmetric, positive definite problems proceed by solving in each iteration, a number of independent systems on subspaces. The convergence of such methods is determined by the spectrum of the sums of orthogonal projections on those subspaces, while the convergence of a related sequential method is determined by the spectrum of the product of complementary projections. We study spectral properties of sums of orthogonal projections and in the case of two projections, characterize the spectrum of the sum completely in terms of the spectrum of the product.
Type of Medium: Electronic Resource
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• 6
Electronic Resource
Chichester [u.a.] : Wiley-Blackwell
International Journal for Numerical Methods in Engineering 29 (1990), S. 1095-1108
ISSN: 0029-5981
Keywords: Engineering ; Engineering General
Source: Wiley InterScience Backfile Collection 1832-2000
Topics: Mathematics , Technology
Notes: This paper presents a flexible approach to domain decomposition preconditioning for problems expressed in terms of local stiffness matrices of subdomains. Evaluation of the preconditioner requires fully parallelizable local computations and the solution of a global auxiliary problem with few variables per subdomain. Convergence factors can be bounded using properties of the local stiffness matrices only. The method is applied to the p-version finite element method for three-dimensional elasticity. We treat each element as a subdomain and compute explicit convergence bounds.
Type of Medium: Electronic Resource
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• 7
Electronic Resource
Chichester : Wiley-Blackwell
Communications in Numerical Methods in Engineering 9 (1993), S. 233-241
ISSN: 1069-8299
Keywords: Engineering ; Engineering General
Source: Wiley InterScience Backfile Collection 1832-2000
Topics: Mathematics , Technology
Notes: The Neumann-Neumann 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 (De Roeck and Le Tallec, 1991). 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 arise in elasticity. A convergence bound independent of the number of subdomains is proved and results of computational tests are reported.
Type of Medium: Electronic Resource
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• 8
Electronic Resource
Chichester [u.a.] : Wiley-Blackwell
International Journal for Numerical Methods in Engineering 38 (1995), S. 3831-3853
ISSN: 0029-5981
Keywords: domain decomposition ; dynamics ; nonlinear ; parallel processing ; Engineering ; Engineering General
Source: Wiley InterScience Backfile Collection 1832-2000
Topics: Mathematics , Technology
Notes: We present a new efficient and scalable domain decomposition method for solving implicitly linear and non-linear time-dependent problems in computational mechanics. The method is derived by adding a coarse problem to the recently proposed transient FETI substructuring algorithm in order to propagate the error globally and accelerate convergence. It is proved that in the limit for large time steps, the new method converges toward the FETI algorithm for time-independent problems. Computational results confirm that the optimal convergence properties of the time-independent FETI method are preserved in the time-dependent case. We employ an iterative scheme for solving efficiently the coarse problem on massively parallel processors, and demonstrate the effective scalability of the new transient FETI method with the large-scale finite element dynamic analysis on the Paragon XP/S and IBM SP2 systems of several diffraction grating finite element structural models. We also show that this new domain decomposition method outperforms the popular direct skyline solver. The coarse problem presented herein is applicable and beneficial to a large class of Lagrange multiplier based substructuring algorithms for time-dependent problems, including the fictitious domain decomposition method.
Type of Medium: Electronic Resource
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• 9
Electronic Resource
New York, NY [u.a.] : Wiley-Blackwell
Communications in Applied Numerical Methods 8 (1992), S. 561-561
ISSN: 0748-8025
Keywords: Engineering ; Engineering General
Source: Wiley InterScience Backfile Collection 1832-2000
Topics: Mathematics , Technology
Type of Medium: Electronic Resource
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• 10
Book
Providence, RI :AMS,
Title: Domain decompsition methods 10 : the 10th international conference on domain decomposition methods, August 10-14, 1997, Boulder, CO.; 218
Publisher: Providence, RI :AMS,
Year of publication: 1998
Pages: 554 S.
Series Statement: Contemporary mathematics 218
Type of Medium: Book
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