On a Parallel Implementation of the BDDC Method and Its Application to the Stokes Problem
01 Jan 2010-
TL;DR: A parallel implementation of the Balancing Domain Decomposition by Constraints (BDDC) method is described that is based on formulation of BDDC with global matrices without explicit coarse problem and successfully applied to several problems of Stokes flow.
Abstract: The Balancing Domain Decomposition by Constraints (BDDC) method is one of the most advanced preconditioners for very large problems with symmetric positive definite matrices that arise from FEM analysis. It is very suitable for massively parallel computers. In presented paper, a parallel implementation of the method is described that is based on formulation of BDDC with global matrices without explicit coarse problem. It is based on the MUMPS parallel solver. The implementation is successfully applied to several problems of Stokes flow and BDDC is shown to be a very promising method also for this class of problems.
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TL;DR: In this paper, the authors combine the two approaches into a new method preserving parallel scalability with increasing number of subdomains and excellent convergence properties, which is one of the most advanced methods of iterative substructuring.
Abstract: The BDDC method [2] is one of the most advanced methods of iterative substructuring. In the case of many substructures, solving the coarse problem exactly becomes a bottleneck. Since the coarse problem has the same structure as the original problem, it is straightforward to apply the method recursively to solve it only approximately. The two-level BDDC analysis has been extended into three-levels in a pioneering work in [14, 15], and into a general multilevel method in [11]. The methods for the adaptive selection of constraints for the two-level BDDC method have been studied in [9, 12]. Here we combine the two approaches into a new method preserving parallel scalability with increasing number of subdomains and excellent convergence properties.
13 citations
TL;DR: The results showed that the parallel SIMPLE algorithm that uses the two‐way finite difference stencil algorithm based on alternate tiling has good data locality, performance, and scalability in the Deepcomp7000 cluster computing environment.
Abstract: Parallel semi-implicit method for pressure-linked equationsSIMPLE algorithm is used to solve the 3-D incompressible pipe flow problem. In this paper, we proposed a novel parallel SIMPLE algorithm that uses the alternate tiling technique. Firstly, a parallel SIMPLE algorithm based on domain decomposition method was established, and the implementation of domain partition and data exchange was presented. Then, we presented serial finite difference stencil algorithm based on alternate tiling. Furthermore, an iteration space parallel two-way finite difference stencil algorithm based on alternate tiling was proposed, introducing the sequence of iterative space tiles as the sequence of execution and using time skewing technique to partition the iteration space, thus to improve the data locality of algorithm. The cache misses and the cost of communication and synchronization are reduced by reordering the tiles of iteration space. Finally, the effectiveness of the two parallel SIMPLE algorithms were compared. The results showed that the parallel SIMPLE algorithm that uses the two-way finite difference stencil algorithm based on alternate tiling has good data locality, performance, and scalability in the Deepcomp7000 cluster computing environment. Copyright © 2013 John Wiley & Sons, Ltd.
5 citations
Cites background from "On a Parallel Implementation of the..."
...The processors exchange data with border at the start of every iteration [9]....
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13 May 2012
TL;DR: A parallel SIMPLE algorithm based on a novel tiling strategy -- alternate tiling, through replacing the original linear system and reordering the iteration space tiles is proposed, which has a good data locality and parallel efficiency in the three-dimensional incompressible pipe flow project.
Abstract: SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm is important in the simulation of steady flows. As the traditional 3-D SIMPLE algorithm is time-consuming, we propose a parallel SIMPLE algorithm based on a novel tiling strategy -- alternate tiling, through replacing the original linear system and reordering the iteration space tiles. The novelty of our parallel algorithm lies in the introduction of the sequence of iteration space tiles as the sequence of execution, the time skewing technique to partition the iteration space, update operations of the grids from two directions alternately, and the improvement of the data locality. The effectiveness of the parallel algorithm and serial model of finite difference stencil algorithm are validated. Numerical experiments on distributed clusters show that the cache misses and the cost of communication and synchronization are reduced by reordering the tiles of iteration space, and the parallel SIMPLE algorithm based on alternate tiling has a good data locality and parallel efficiency in the three-dimensional incompressible pipe flow project.
1 citations
Cites methods from "On a Parallel Implementation of the..."
...Sistek [9] implemented a parallel version of the BDDC (Balancing Domain Decomposition based on Constraints) method and verified its performance in application of solving the Stokes problem....
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References
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Book•
23 Nov 2011TL;DR: Variational Formulations and Finite Element Methods for Elliptic Problems, Incompressible Materials and Flow Problems, and Other Applications.
Abstract: Variational Formulations and Finite Element Methods. Approximation of Saddle Point Problems. Function Spaces and Finite Element Approximations. Various Examples. Complements on Mixed Methods for Elliptic Problems. Incompressible Materials and Flow Problems. Other Applications.
5,030 citations
"On a Parallel Implementation of the..." refers background or methods in this paper
...[2]) used in our computations, we were not able to obtain contributive results by that method....
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...For the case of linear elasticity, a(u, v) is a symmetric positive definite bilinear form on U×U , while for the Stokes problem, it is symmetric indefinite [2, 5]....
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01 Jan 2005
TL;DR: The purpose of this text is to offer a comprehensive and self-contained presentation of some of the most successful and popular domain decomposition preconditioners for finite and spectral element approximations of partial differential equations.
Abstract: The purpose of this text is to offer a comprehensive and self-contained presentation of some of the most successful and popular domain decomposition preconditioners for finite and spectral element approximations of partial differential equations. Strong emphasis is placed on both algorithmic and mathematical aspects. Some important methods such FETI and balancing Neumann-Neumann methods and algorithms for spectral element methods, not treated previously in any monograph, are covered in detail.
Winner of the 2005 Award for Excellence in Professional and Scholarly Publishing - Mathematics/Statistics - of the Association of American Publishers
2,313 citations
"On a Parallel Implementation of the..." refers background in this paper
...The need of first-rate preconditioners tailored to the solved problem, which can be implemented in parallel, gave rise to the field of domain decomposition methods [12]....
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1,011 citations
TL;DR: In this paper, a new parallel distributed memory multifrontal approach is described to handle numerical pivoting efficiently, a parallel asynchronous algorithm with dynamic scheduling of the computing tasks has been developed.
940 citations
TL;DR: A preconditioner for substructuring based on constrained energy minimization concepts is presented and offers a straightforward approach for the iterative solution of second- and fourth-order structural mechanics problems.
Abstract: A preconditioner for substructuring based on constrained energy minimization concepts is presented. The preconditioner is applicable to both structured and unstructured meshes and offers a straightforward approach for the iterative solution of second- and fourth-order structural mechanics problems. The approach involves constraints associated with disjoint sets of nodes on substructure boundaries. These constraints provide the means for preconditioning at both the substructure and global levels. Numerical examples are presented that demonstrate the good performance of the method in terms of iterations, compute time, and condition numbers of the preconditioned equations.
405 citations
"On a Parallel Implementation of the..." refers methods in this paper
...It was introduced by Dohrmann [4] in 2003 and the theory was developed by Mandel and Dohrmann in [8]....
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...The Balancing Domain Decomposition based on Constraints (BDDC) [4, 9] is one of the most advanced preconditioners of this class derived for symmetric positive definite problems....
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