A class of two‐stage iterative methods for systems of weakly nonlinear equations |
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Authors: | Bai Zhong‐Zhi |
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Institution: | (1) State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, P.O. Box 2719, Beijing, 100080, P.R. China |
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Abstract: | The discretizations of many differential equations by the finite difference or the finite element methods can often result
in a class of system of weakly nonlinear equations. In this paper, by applying the two-tage iteration technique and in accordance
with the special properties of this weakly nonlinear system, we first propose a general two-tage iterative method through
the two-tage splitting of the system matrix. Then, by applying the accelerated overrelaxation (AOR) technique of the linear
iterative methods, we present a two-tage AOR method, which particularly uses the AOR iteration as the inner iteration and
is substantially a relaxed variant of the afore-presented method. For these two classes of methods, we establish their local
convergence theories, and precisely estimate their asymptotic convergence factors under some suitable assumptions when the
involved nonlinear mapping is only B-differentiable. When the system matrix is either a monotone matrix or an H-matrix, and
the nonlinear mapping is a P-bounded mapping, we thoroughly set up the global convergence theories of these new methods. Moreover,
under the assumptions that the system matrix is monotone and the nonlinear mapping is isotone, we discuss the monotone convergence
properties of the new two-tage iteration methods, and investigate the influence of the matrix splittings as well as the relaxation
parameters on the convergence behaviours of these methods. Numerical computations show that our new methods are feasible and
efficient for solving the system of weakly nonlinear equations.
This revised version was published online in June 2006 with corrections to the Cover Date. |
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Keywords: | system of weakly nonlinear equations two-tage iteration relaxation technique convergence theory convergence rate 65H10 65W05 |
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