Face-based selection of corners in 3D substructuring

Šístek, Jakub; Čertíková, Marta; Burda, Pavel; Novotný, Jaroslav

doi:10.1016/j.matcom.2011.06.007

Cited by 19 publications

(28 citation statements)

References 20 publications

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“…In two-dimensional problems, G is classified as an edge if |N (G)| = 2 and |G| > 1 and a corner if |G| = 1. A similar classification was introduced in [37,32,48].…”

Section: Preconditioner Formulationmentioning

confidence: 99%

“…Attempts to relax this assumption have been successful only in two dimensions (see [35]). However, BDDC and FETI-DP, whose analysis relies on Lemma 4.2, work well in practice for more general partitions, e.g., those generated by graph partitioners (see [24,33,34,48,7]). …”

Section: Convergence Analysismentioning

confidence: 99%

“…For structured meshes and regular partitions, algorithms to find such constraints are simple, but it would become much more complicated when considering unstructured meshes of complex three-dimensional geometries and partitions resulting from parallel mesh partitioners, e.g., ParMETIS [27] or PT-Scotch [16]. In order to have well-posedness, different corner selection algorithms have been considered [18,40,37,48]. The idea behind these algorithms is to introduce enough corner constraints such that the partially subassembled problems are provably well-posed even with corner constraints only.…”

mentioning

confidence: 99%

“…Based on our experience, the implementation of this type of algorithm is an involved and timeconsuming task which depends on the physical problem to be solved and also the type of FE formulation being used. For example, the implementation of the corner selection algorithm from [48] in the scientific software FEMPAR [4] has more than 1,500 lines of code. Furthermore, the situation becomes far more complicated when subdomains are disconnected, or connected only by corners or edges.…”

mentioning

confidence: 99%

“…1 As far as we know, these situations have not been fully considered when formulating the corner selection algorithms [40,18,48], thus leading to the breakdown of the BDDC method. It is certainly possible to deal with disconnected subdomains, but it will increase the findings.…”

mentioning

confidence: 99%

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Balancing Domain Decomposition by Constraints and Perturbation

Badia¹,

Nguyen²

2016

SIAM J. Numer. Anal.

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Abstract. In this paper, we formulate and analyze a perturbed formulation of the balancing domain decomposition by constraints (BDDC) method. We prove that the perturbed BDDC has the same polylogarithmic bound for the condition number as the standard formulation. Two types of properly scaled zero-order perturbations are considered: one uses a mass matrix, and the other uses a Robin-type boundary condition, i.e, a mass matrix on the interface. With perturbation, the wellposedness of the local Neumann problems and the global coarse problem is automatically guaranteed, and coarse degrees of freedom can be defined only for convergence purposes but not well-posedness. This allows a much simpler implementation as no complicated corner selection algorithm is needed. Minimal coarse spaces using only face or edge constraints can also be considered. They are very useful in extreme scale calculations where the coarse problem is usually the bottleneck that can jeopardize scalability. The perturbation also adds extra robustness as the perturbed formulation works even when the constraints fail to eliminate a small number of subdomain rigid body modes from the standard BDDC space. This is extremely important when solving problems on unstructured meshes partitioned by automatic graph partitioners since arbitrary disconnected subdomains are possible. Numerical results are provided to support the theoretical findings.Key words. BDDC, preconditioner, coarse space, parallel solver, scalability AMS subject classifications. 65N55, 65N22, 65F08 DOI. 10.1137/15M10456481. Introduction. The development of highly scalable linear solvers for the solution of large scale linear systems arising from the finite element (FE) discretization of second-order elliptic problems on distributed-memory machines is of great importance in many applications. In this work, we consider nonoverlapping domain decomposition (DD) methods [49], which take advantage of the partition of the FE mesh into submeshes to define effective preconditioners that can exploit large levels of concurrency. In particular, we focus on a variant of the balancing DD by constraints (BDDC) method. The BDDC method was first introduced in 2003 by Dohrmann [18]. It can be regarded as an improved version of the balancing domain decomposition (BDD) method by Mandel [43]. It also has a very close connection with the dual primal finite element tearing and interconnecting (FETI-DP) method [25,24]. In fact, the eigenvalues of the preconditioned systems in the two approaches are almost identical [44,41,15]. The BDDC method is particularly well suited for extreme scale simulations, since it allows for a very aggressive coarsening, the computations at different levels can be computed in parallel, the subdomain problems can be solved inexactly

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“…In two-dimensional problems, G is classified as an edge if |N (G)| = 2 and |G| > 1 and a corner if |G| = 1. A similar classification was introduced in [37,32,48].…”

Section: Preconditioner Formulationmentioning

confidence: 99%

Section: Convergence Analysismentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Balancing Domain Decomposition by Constraints and Perturbation

Badia¹,

Nguyen²

2016

SIAM J. Numer. Anal.

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BDDC for mixed‐hybrid formulation of flow in porous media with combined mesh dimensions

Šístek

Březina

Sousedík

2015

Numerical Linear Algebra App

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SUMMARYWe extend the Balancing Domain Decomposition by Constraints (BDDC) method to flows in porous media discretised by mixed-hybrid finite elements with combined mesh dimensions. Such discretisations appear when major geological fractures are modelled by 1D or 2D elements inside three-dimensional domains. In this set-up, the global problem as well as the substructure problems have a symmetric saddle-point structure, containing a 'penalty' block due to the combination of meshes. We show that the problem can be reduced by means of iterative substructuring to an interface problem, which is symmetric and positive definite. The interface problem can thus be solved by conjugate gradients with the BDDC method as a preconditioner. A parallel implementation of this algorithm is incorporated into an existing software package for subsurface flow simulations. We study the performance of the iterative solver on several academic and real-world problems. Numerical experiments illustrate its efficiency and scalability. Preprint version.

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Enhanced balancing Neumann–Neumann preconditioning in computational fluid and solid mechanics

Badia

Martı́n

Principe

2013

Numerical Meth Engineering

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In this work, we propose an enhanced implementation of balancing Neumann-Neumann (BNN) preconditioning together with a detailed numerical comparison against the balancing domain decomposition by constraints (BDDC) preconditioner. As model problems, we consider the Poisson and linear elasticity problems. On one hand, we propose a novel way to deal with singular matrices and pseudo-inverses appearing in local solvers. It is based on a kernel identification strategy that allows us to efficiently compute the action of the pseudo-inverse via local indefinite solvers. We further show how, identifying a minimum set of degrees of freedom to be fixed, an equivalent definite system can be solved instead, even in the elastic case. On the other hand, we propose a simple implementation of the algorithm that reduces the number of Dirichlet solvers to only one per iteration, leading to similar computational cost as additive methods. After these improvements of the BNN preconditioned conjugate gradient algorithm, we compare its performance against that of the BDDC preconditioners on a pair of large-scale distributed-memory platforms. The enhanced BNN method is a competitive preconditioner for three-dimensional Poisson and elasticity problems and outperforms the BDDC method in many cases. of these functions, denoted by I i Dir.& į /, which consists on assigning the DoFs of Dirichlet type to zero. We consider the coarse space ‡ An intrusive approach requires the sources of the direct solver software. As an example, none of these solvers provide the sources for modifications. § As an example, the algorithms in [6,7] require knowing the number of the last nodes that can be placed on a line. ¶ To the best of our knowledge, an additive BNN method has not been proposed so far (Section 4.3)

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Face-based selection of corners in 3D substructuring

Cited by 19 publications

References 20 publications

Balancing Domain Decomposition by Constraints and Perturbation

Balancing Domain Decomposition by Constraints and Perturbation

BDDC for mixed‐hybrid formulation of flow in porous media with combined mesh dimensions

Enhanced balancing Neumann–Neumann preconditioning in computational fluid and solid mechanics

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