Domain Decomposition Methods and Deflated Krylov Subspace Iterations
نویسندگان
چکیده
The balancing Neumann-Neumann (BNN) and the additive coarse grid correction (BPS) preconditioner are fast and successful preconditioners within domain decomposition methods for solving partial differential equations. For certain elliptic problems these preconditioners lead to condition numbers which are independent of the mesh sizes and are independent of jumps in the coefficients (BNN). Here we give an algebraic formulation of these preconditioners. This formulation allows a comparison with another solution or preconditioning technique the deflation technique. By giving a detailed introduction into the deflation technique we establish analogies between the balancing, the additive coarse grid correction and the deflation technique. We prove that the effective condition number of the deflated preconditioned system is always, i.e. for all deflation vectors and all restrictions and prolongations, below the condition number of the system preconditioned by the balancing preconditioner and the coarse grid correction preconditoner. This implies that the conjugate gradient method applied to the deflated preconditioned system is expected to converge always faster than the conjugate gradient method applied to the system preconditioned by the coarse grid correction or balancing. Moreover, we prove that the A-norm of the errors of the iterates built by the deflation preconditioner is always below the A-norm of the errors of the iterates built by the balancing preconditioner. Numerical results for porous media flows emphasize the theoretical results.
منابع مشابه
Domain Decomposition Methods for a Complementarity Problem∗
We introduce a family of parallel Newton-Krylov-Schwarz methods for solving complementarity problems. The methods are based on a smoothed grid sequencing method, a semismooth inexact Newton method, and a twogrid restricted overlapping Schwarz preconditioner. We show numerically that such an approach is highly scalable in the sense that the number of Newton iterations and the number of linear it...
متن کاملSpectral Deflation in Krylov Solvers: a Theory of Coordinate Space Based Methods
For the iterative solution of large sparse linear systems we develop a theory for a family of augmented and deflated Krylov space solvers that are coordinate based in the sense that the given problem is transformed into one that is formulated in terms of the coordinates with respect to the augmented bases of the Krylov subspaces. Except for the augmentation, the basis is as usual generated by a...
متن کاملA Framework for Deflated and Augmented Krylov Subspace Methods
We consider deflation and augmentation techniques for accelerating the convergence of Krylov subspace methods for the solution of nonsingular linear algebraic systems. Despite some formal similarity, the two techniques are conceptually different from preconditioning. Deflation (in the sense the term is used here) “removes” certain parts from the operator making it singular, while augmentation a...
متن کاملScalable Parallel Domain Decomposition Methods for Numerical Simulation of PDEs
This paper is concerned about scalable parallel domain decomposition methods for numerical simulation of PDEs. First, one level and two level scalable parallel domain decomposition methods which can be used to solve different equations, are introduced in detail, and then we explain Krylov subspace accelerator technique used to improve the convergence of the methods. Last, the results of some nu...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2006