Balancing Neumann-Neumann Methods for Mixed Approximations of Linear Elasticity

Goldfeld, Paulo; Pavarino, Luca F.; Widlund, Olof B.

doi:10.1007/978-3-642-56118-4_4

Cited by 5 publications

(7 citation statements)

References 25 publications

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“…The algorithm has been implemented by the first author in C, using the PETSc library. We report on results for compressible/almost-incompressible elasticity only, although similar results have been obtained for incompressible elasticity (and also Stokes and generalized Stokes equations), see [17].…”

Section: Parallel Results For Elasticity and Q 2 − Q 0 Mixed Finite Ementioning

confidence: 99%

“…We will use a balancing Neumann-Neumann domain decomposition method and we note that this is an extension of our recent work on incompressible Stokes's equations [30]; the main results of this paper have also been reported without proofs in a conference paper [17]. The Stokes and elasticity problems in mixed form have much in common but both the algorithm and analysis have to be modified, in particular, when considering large variations in the material properties.…”

Section: Introductionmentioning

confidence: 99%

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Balancing Neumann-Neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity

Goldfeld

Pavarino

Widlund

2003

Numerische Mathematik

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Balancing Neumann-Neumann methods are extented to mixed formulations of the linear elasticity system with discontinuous coefficients, discretized with mixed finite or spectral elements with discontinuous pressures. These domain decomposition methods implicitly eliminate the degrees of freedom associated with the interior of each subdomain and solve iteratively the resulting saddle point Schur complement using a hybrid preconditioner based on a coarse mixed elasticity problem and local mixed elasticity problems with natural and essential boundary conditions. A polylogarithmic bound in the local number of degrees of freedom is proven for the condition number of the preconditioned operator in the constant coefficient case. Parallel and serial numerical experiments confirm the theoretical results, indicate that they still hold for systems with discontinuous coefficients, and show that our algorithm is scalable, parallel, and robust with respect to material 284 P. Goldfeld et al. heterogeneities. The results on heterogeneous general problems are also supported in part by our theory. Classification (1991): 65N55, 65N30, 65N35, 65F10, 65Y05 Mathematics Subject

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Section: Parallel Results For Elasticity and Q 2 − Q 0 Mixed Finite Ementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Balancing Neumann-Neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity

Goldfeld

Pavarino

Widlund

2003

Numerische Mathematik

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“…Let L 1 k ( ) be the finite element space of polynomials of degree k. We define scalar-and vector-valued finite element spaces (8) and (12), respectively. Here, we associate a pair [ 0 , k] with ∈ Y as (11). Let…”

Section: Assumption 3 a Union Of Partitionsmentioning

confidence: 99%

“…They were first developed for problems of structural mechanics and their extension to other problems has been a topic of development of domain decomposition methods, e.g. balancing Neumann-Neumann methods for the Stokes equations [10] and elasticity problem for almost incompressible material [11], and a dual-primal FETI method for the Stokes/Navier-Stokes equations [12]. In these methods, discontinuous pressure approximation is used to treat incompressibility.…”

Section: Introductionmentioning

confidence: 99%

Finite element matrices in congruent subdomains and their effective use for large-scale computations

Suzuki

Tabata

2005

Int. J. Numer. Meth. Engng

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SUMMARYThe structure of finite element matrices in congruent subdomains is studied. When a domain has a form of symmetries and/or periodicities, it is decomposed into a union of congruent subdomains, each of which is an image of a reference subdomain by an affine transformation with an orthogonal matrix whose components consist of −1, 0, and 1. Stiffness matrices in subdomains are expressed by one in the reference subdomain with renumbering indices and changing signs corresponding to the orthogonal matrices. The memory requirements for a finite element solver are reduced by the domain decomposition, which is useful in large-scale computations. Reducing rates of memory requirements to store matrices are reported with examples of domains. Both applicability and limitations of the algorithm are discussed with an application to the Earth's mantle convection problem.

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“…This method was applied to stationary Stokes problems by Pavarino and Widlund (2) and Goldfield (3) and has been extended to a class of saddle point problems and the mixed formulations of linear elasticity by Goldfeld, . et al (4) . For stationary Stokes problems,…”

Section: Introductionmentioning

confidence: 97%

Balancing Domain Decomposition for Non-stationary Incompressible Flow Problems Using a Characteristic-curve Method

Yao

Kitamura

Notsu

et al. 2010

JCST

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Numerical solutions for large scale models face the problem of poor convergence and for non-stationary problems the situation becomes worse. To speed up the convergence, an algorithm to perform the Balancing Domain Decomposition (BDD) preconditioning for non-stationary incompressible flow problems is constructed. The problem caused by the presence of nonlinear convection term in the non-stationary Navier-Stokes equations, which complicates the solving of the equations numerically, is solved by a characteristic-curve method. The method is advantageous as it renders the matrix for linear equations symmetric, thus enabling the Conjugate Gradient (CG) method to be used to solve the interface problem of the Domain Decomposition Method (DDM). The related algorithms are all implemented in parallel by the Hierarchical Domain Decomposition Method (HDDM) and the program has the capability of solving non-stationary problems with up to 10 million degrees of freedom (DOF). The BDD algorithm together with characteristic-curve method is applied to a benchmark problem, a three-dimensional driven cavity flow model, and the results are compared with available solutions.

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Balancing Neumann-Neumann Methods for Mixed Approximations of Linear Elasticity

Cited by 5 publications

References 25 publications

Balancing Neumann-Neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity

Balancing Neumann-Neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity

Finite element matrices in congruent subdomains and their effective use for large-scale computations

Balancing Domain Decomposition for Non-stationary Incompressible Flow Problems Using a Characteristic-curve Method

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