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Additive Schwarz Preconditioner (additive + schwarz_preconditioner)

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Engineering60%

Selected Abstracts

A note on the mesh independence of convergence bounds for additive Schwarz preconditioned GMRES

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 6 2008
Xiuhong Du
Abstract Additive Schwarz preconditioners, when including a coarse grid correction, are said to be optimal for certain discretized partial differential equations, in the sense that bounds on the convergence of iterative methods are independent of the mesh size h. Cai and Zou (Numer. Linear Algebra Appl. 2002; 9:379,397) showed with a one-dimensional example that in the absence of a coarse grid correction the usual GMRES bound has a factor of the order of . In this paper we consider the same example and show that for that example the behavior of the method is not well represented by the above-mentioned bound: We use an a posteriori bound for GMRES from (SIAM Rev. 2005; 47:247,272) and show that for that example a relevant factor is bounded by a constant. Furthermore, for a sequence of meshes, the convergence curves for that one-dimensional example, and for several two-dimensional model problems, are very close to each other; thus, the number of preconditioned GMRES iterations needed for convergence for a prescribed tolerance remains almost constant. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Application of the additive Schwarz method to large scale Poisson problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2004
K. M. Singh
Abstract This paper presents an application of the additive Schwarz method to large scale Poisson problems on parallel computers. Domain decomposition in rectangular blocks with matching grids on a structured rectangular mesh has been used together with a stepwise approximation to approximate sloping sides and complicated geometric features. A seven-point stencil based on central difference scheme has been used for the discretization of the Laplacian for both interior and boundary grid points, and this results in a symmetric linear algebraic system for any type of boundary conditions. The preconditioned conjugate gradient method has been used as an accelerator for the additive Schwarz method, and three different methods have been assessed for the solution of subdomain problems. Numerical experiments have been performed to determine the most suitable set of subdomain solvers and the optimal accuracy of subdomain solutions; to assess the effect of different decompositions of the problem domain; and to evaluate the parallel performance of the additive Schwarz preconditioner. Application to a practical problem involving complicated geometry is presented which establishes the efficiency and robustness of the method. Copyright © 2004 John Wiley & Sons, Ltd. [source]

A hybrid domain decomposition method based on aggregation

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 4 2004
*Article first published online: 19 APR 200, Yu. Vassilevski
Abstract A new two-level black-box preconditioner based on the hybrid domain decomposition technique is proposed and studied. The preconditioner is a combination of an additive Schwarz preconditioner and a special smoother. The smoother removes dependence of the condition number on the number of subdomains and variations of the diffusion coefficient and leaves minor sensitivity to the problem size. The algorithm is parallel and pure algebraic which makes it a convenient framework for the construction parallel black-box preconditioners on unstructured meshes. Copyright © 2004 John Wiley & Sons, Ltd. [source]

Non-linear additive Schwarz preconditioners and application in computational fluid dynamics

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2002
Xiao-Chuan Cai
Abstract The focus of this paper is on the numerical solution of large sparse non-linear systems of algebraic equations on parallel computers. Such non-linear systems often arise from the discretization of non-linear partial differential equations, such as the Navier,Stokes equations for fluid flows, using finite element or finite difference methods. A traditional inexact Newton method, applied directly to the discretized system, does not work well when the non-linearities in the algebraic system become unbalanced. In this paper, we study some preconditioned inexact Newton algorithms, including the single-level and multilevel non-linear additive Schwarz preconditioners. Some results for solving the high Reynolds number incompressible Navier,Stokes equations are reported. Copyright © 2002 John Wiley & Sons, Ltd. [source]