Domain decomposition for the incompressible Navier-Stokes equations: solving subdomain problems accurately and inaccurately

Brakkee, E.; Vuik, C.; Wesseling, Pieter

doi:10.1002/(sici)1097-0363(19980615)26:10<1217::aid-fld693>3.0.co;2-m

Cited by 23 publications

(16 citation statements)

References 39 publications

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“…This research is a continuation of our work presented in [2,3,9,25,27]. For an overview of the literature of related parallel methods we refer to [9,27].…”

Section: Introductionmentioning

confidence: 90%

Coarse grid acceleration of a parallel block preconditioner

Vuik

Frank

2001

Future Generation Computer Systems

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“…This research is a continuation of our work presented in [2,3,9,25,27]. For an overview of the literature of related parallel methods we refer to [9,27].…”

Section: Introductionmentioning

confidence: 90%

Coarse grid acceleration of a parallel block preconditioner

Vuik

Frank

2001

Future Generation Computer Systems

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“…The linear subsystems of equations can be solved by either direct or iterative methods; the latter are typically more efficient, see for instance [10,[12][13][14][15]. However, we use both approaches to find the solution of the subsystems.…”

Section: Subsystems Solution: Direct and Iterative Methodsmentioning

confidence: 99%

A multiplicative Schwarz method with active subdomains for transient convection–diffusion problems

Sandoval

Ferran

2009

Numer Methods Biomed Eng

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upwind Petrov-Galerkin, Galerkin/Least-Squares (GLS) and sub-grid scale (SGS). Least-squares stabilization, on the contrary, leads to a symmetric matrix. Hence, we should solve thousands of linear algebraic systemswhere the matrix A is large, sparse, symmetric, positive definite and constant at each time-step n. A finite element discretization described in [5] is applied. Owing to the formation of an abrupt front, the global solution varies only in a small part of the complete domain at each time-step. Our proposal is to exploit this fact by combining a domain decomposition method with the activation and inactivation of subdomains depending on how the hydrocarbon concentration front moves.We decided to use the multiplicative Schwarz method (MSM) because it converges faster than the additive Schwarz method [6], and its convergence rate depends on both the overlap size and the number of subdomains [7,8]. We want to emphasize that the MSM is used directly as a block Gauss-Seidel solver [9] rather than as a preconditioner.Since it is important to order the subdomains (and nodes) in the flow direction [6, 10], the nodes of the mesh are reordered following the typical block-interface structure of many activated-carbon filters, see [5]. Therefore, a block-tridiagonal global matrix is obtained. In addition, for each block, the reverse Cuthill-McKee algorithm [11] is used to reduce the bandwidth of the matrix.This article is organized as follows. In Section 2, we define the subdomains that are used in the MSM. In Section 3, we introduce the appropriate criteria to activate or inactivate subdomains. In Section 4, we present our proposal: the MSM with active subdomains. In Section 5, we show some numerical experiments illustrating the performance of the new algorithm, as well as some comparisons with the classical direct and iterative solvers. We close the paper with some concluding remarks. DEFINITION OF SUBDOMAINS FOR THE MSM

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“…In (Brakkee et al, 1997), interesting results were obtained with inaccurate subdomain solutions. Applying the same strategy to our application would require used a method that can handle a variable preconditioner, such as the flexible CG method (Axelsson, 1994).…”

Section: Future Workmentioning

confidence: 99%

Two implementations of the preconditioned conjugate gradient method on heterogeneous computing grids

Collignon¹,

Gijzen²

2010

International Journal of Applied Mathematics and Computer Science

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Efficient iterative solution of large linear systems on grid computers is a complex problem. The induced heterogeneity and volatile nature of the aggregated computational resources present numerous algorithmic challenges. This paper describes a case study regarding iterative solution of large sparse linear systems on grid computers within the software constraints of the grid middleware GridSolve and within the algorithmic constraints of preconditioned Conjugate Gradient (CG) type methods. We identify the various bottlenecks induced by the middleware and the iterative algorithm. We consider the standard CG algorithm of Hestenes and Stiefel, and as an alternative the Chronopoulos/Gear variant, a formulation that is potentially better suited for grid computing since it requires only one synchronisation point per iteration, instead of two for standard CG. In addition, we improve the computation-to-communication ratio by maximising the work in the preconditioner. In addition to these algorithmic improvements, we also try to minimise the communication overhead within the communication model currently used by the GridSolve middleware. We present numerical experiments on 3D bubbly flow problems using heterogeneous computing hardware that show lower computing times and better speed-up for the Chronopoulos/Gear variant of conjugate gradients. Finally, we suggest extensions to both the iterative algorithm and the middleware for improving granularity.

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Domain decomposition for the incompressible Navier-Stokes equations: solving subdomain problems accurately and inaccurately

Cited by 23 publications

References 39 publications

Coarse grid acceleration of a parallel block preconditioner

Coarse grid acceleration of a parallel block preconditioner

A multiplicative Schwarz method with active subdomains for transient convection–diffusion problems

Two implementations of the preconditioned conjugate gradient method on heterogeneous computing grids

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