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On iterative solution of Helmholtz’ equations on a sphere
Abstract
The efficiency of incomplete factorization with conjugate-gradients acceleration is studied for difference Helmholtz’ equations on a sphere with a special numeration of the grid nodes. Economical implementations of the methods of cyclic and block-cyclic reduction are given. The converge rate of the method is estimated and numerical examples are given.
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The paper considers some versions of domain decomposition methods for solving five-point mesh equations resulting from approximations of two-dimensional boundary value problems on rectangular meshes. We suggest the conjugate gradient iterative method with a preconditioner in the form of an incomplete factorization of the Schur complement and derive upper estimates of its convergence rate for the case where the original domain is decomposed into two subdomains.
The paper analyses explicit and implicit algorithms for incomplete factorization. The structure of preconditioning matrices and properties of their entries are considered as well as the conditions for the correctness of methods and for the convergence of iterative processes. The convergence rate estimates for iterations are given, and the results of numerical experiments are discussed.