The mosaic of high performance domain Decomposition Methods for Structural Mechanics: Formulation, interrelation and numerical efficiency of primal and dual methods

Fragakis, Yannis; Papadrakakis, Manolis

doi:10.1016/s0045-7825(03)00374-8

Cited by 93 publications

(98 citation statements)

References 20 publications

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“…We then provide a new, short proof of an important result, due to Mandel, Dohrmann, and Tezaur [14], which shows that the eigenvalues of a pair of FETI-DP and BDDC algorithms, with the same sets of primal constraints, are the same. We note that this fact was first observed experimentally by Fragakis and Papadrakakis [6]; these authors also discussed primal iterative substructuring methods which are close counterparts to FETI algorithms. A consequence of the result in [6,14] is that there is no real difference between the convergence rates of a FETI-DP and a BDDC algorithm with the same set of primal constraints.…”

Section: Introductionsupporting

confidence: 60%

“…We note that this fact was first observed experimentally by Fragakis and Papadrakakis [6]; these authors also discussed primal iterative substructuring methods which are close counterparts to FETI algorithms. A consequence of the result in [6,14] is that there is no real difference between the convergence rates of a FETI-DP and a BDDC algorithm with the same set of primal constraints. The choice of algorithm can therefore be based on other considerations; we note, in particular, that it appears easier to modify BDDC than FETI-DP when introducing additional levels; cf.…”

Section: Introductionsupporting

confidence: 60%

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FETI-DP, BDDC, and block Cholesky methods

Widlund

2006

Int. J. Numer. Meth. Engng

190

191

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Two popular non-overlapping domain decomposition methods, the FETI-DP and BDDC algorithms, are reformulated using Block Cholesky factorizations, an approach which can provide a useful framework for the design of domain decomposition algorithms for solving symmetric positive definite linear system of equations. Instead of introducing Lagrange multipliers to enforce the coarse level, primal continuity constraints in these algorithms, a change of variables is used such that each primal constraint corresponds to an explicit degree of freedom. With the new formulations of these algorithms, a simplified proof is provided that the spectra of a pair of FETI-DP and BDDC algorithms, with the same set of primal constraints, are the same. Results of numerical experiments also confirm this result.

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Section: Introductionsupporting

confidence: 60%

Section: Introductionsupporting

confidence: 60%

FETI-DP, BDDC, and block Cholesky methods

Widlund

2006

Int. J. Numer. Meth. Engng

190

191

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“…The balancing domain decomposition methods by constraints (BDDC) were introduced by Dohrmann [6], see also [9] and [5] for related algorithms. These algorithms originally were designed for the symmetric, positive definite systems.…”

Section: Introductionmentioning

confidence: 99%

BDDC for Nonsymmetric Positive Definite and Symmetric Indefinite Problems

Tu¹,

Li²

2009

Lecture Notes in Computational Science and Engineering

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Summary. The balancing domain decomposition methods by constraints are extended to solving both nonsymmetric, positive definite and symmetric, indefinite linear systems. In both cases, certain nonstandard primal constraints are included in the coarse problems of BDDC algorithms to accelerate the convergence. Under the assumption that the subdomain size is small enough, a convergence rate estimate for the GMRES iteration is established that the rate is independent of the number of subdomains and depends only slightly on the subdomain problem size. Numerical experiments for several two-dimensional examples illustrate the fast convergence of the proposed algorithms.

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“…[10,Section 6.2] where references to earlier work can also be found. It has also been established that the preconditioned operators of a pair of BDDC and FETI-DP algorithms, with the same primal constraints, have the same nonzero eigenvalues for positive definite elliptic problems; see [9,3,7].…”

Section: Introductionmentioning

confidence: 99%

A BDDC Preconditioner for Saddle Point Problems

Widlund

Lecture Notes in Computational Science and Engineering

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Summary. The purpose of this paper is to extend the BDDC (balancing domain decomposition by constraints) algorithm to saddle point problems that arise when mixed finite element methods are used to approximate the system of incompressible Stokes equations. The BDDC algorithms are defined in terms of a set of primal continuity constraints, which are enforced across the interface between the subdomains, and which provide a coarse space component of the preconditioner. Sets of such constraints are identified for which bounds on the rate of convergence can be established that are just as strong as previously known bounds for the elliptic case. The preconditioned operator is positive definite and a conjugate gradient method can be used. A close connection is also established between the BDDC and FETI-DP algorithms for the Stokes case.

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The mosaic of high performance domain Decomposition Methods for Structural Mechanics: Formulation, interrelation and numerical efficiency of primal and dual methods

Cited by 93 publications

References 20 publications

FETI-DP, BDDC, and block Cholesky methods

FETI-DP, BDDC, and block Cholesky methods

BDDC for Nonsymmetric Positive Definite and Symmetric Indefinite Problems

A BDDC Preconditioner for Saddle Point Problems

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