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Two-Grid Methods for Hermitian positive definite linear systems connected with an order relation

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Abstract

Given a multigrid procedure for linear systems with coefficient matrices \(A_n,\) we discuss the optimality of a related multigrid procedure with the same smoother and the same projector, when applied to properly related algebraic problems with coefficient matrices \(B_n\): we assume that both \(A_n\) and \(B_n\) are Hermitian positive definite with \(A_n\le \vartheta B_n,\) for some positive \(\vartheta \) independent of \(n.\) In this context we prove the Two-Grid Method optimality. We apply this elementary strategy for designing a multigrid solution for modifications of multilevel structured linear systems, in which the Hermitian positive definite coefficient matrix is banded in a multilevel sense. As structured matrices, Toeplitz, circulants, Hartley, sine (\(\tau \) class) and cosine algebras are considered. In such a way, several linear systems arising from the approximation of integro–differential equations with various boundary conditions can be efficiently solved in linear time (with respect to the size of the algebraic problem). Some numerical experiments are presented and discussed, both with respect to Two-Grid and multigrid procedures.

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Acknowledgments

The authors thank the Referees for their very pertinent and useful remarks.

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Authors and Affiliations

  1. Dipartimento di Scienza e alta Tecnologia, Università dell’Insubria—sede di Como, via Valleggio, 11, 22100 , Como, Italy

    Stefano Serra-Capizzano

  2. Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Via Cozzi, 53, 20125 , Milano, Italy

    Cristina Tablino-Possio

Authors
  1. Stefano Serra-Capizzano
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  2. Cristina Tablino-Possio
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Correspondence to Cristina Tablino-Possio.

Additional information

The work of the authors was partially supported by MIUR 20083KLJEZ.

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Serra-Capizzano, S., Tablino-Possio, C. Two-Grid Methods for Hermitian positive definite linear systems connected with an order relation. Calcolo 51, 261–285 (2014). https://doi.org/10.1007/s10092-013-0081-9

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  • DOI: https://doi.org/10.1007/s10092-013-0081-9

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