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Convergence analysis of a balancing domain decomposition method for solving a class of indefinite linear systems
Author(s) -
Li Jing,
Tu Xuemin
Publication year - 2009
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.639
Subject(s) - mathematics , domain decomposition methods , finite element method , discretization , convergence (economics) , partial differential equation , helmholtz equation , eigenvalues and eigenvectors , system of linear equations , positive definite matrix , boundary value problem , linear system , rate of convergence , mathematical analysis , computer science , physics , economics , thermodynamics , economic growth , computer network , channel (broadcasting) , quantum mechanics
A variant of balancing domain decomposition method by constraints (BDDC) is proposed for solving a class of indefinite systems of linear equations of the form ( K −σ 2 M ) u = f , which arise from solving eigenvalue problems when an inverse shifted method is used and also from the finite element discretization of Helmholtz equations. Here, both K and M are symmetric positive definite. The proposed BDDC method is closely related to the previous dual–primal finite element tearing and interconnecting method (FETI‐DP) for solving this type of problems ( Appl. Numer. Math. 2005; 54 :150–166), where a coarse level problem containing certain free‐space solutions of the inherent homogeneous partial differential equation is used in the algorithm to accelerate the convergence. Under the condition that the diameters of the subdomains are small enough, the convergence rate of the proposed algorithm is established, which depends polylogarithmically on the dimension of the individual subdomain problems and which improves with a decrease of the subdomain diameters. These results are supported by numerical experiments of solving a two‐dimensional problem. Copyright © 2009 John Wiley & Sons, Ltd.