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Domain Decomposition Methods in Science and Engineering pp 201–208Cite as

Improving Robustness and Parallel Scalability of Newton Method Through Nonlinear Preconditioning

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Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 40))

Summary

Inexact Newton method with backtracking is one of the most popular techniques for solving large sparse nonlinear systems of equations. The method is easy to implement, and converges well for many practical problems. However, the method is not robust. More precisely speaking, the convergence may stagnate for no obvious reason. In this paper, we extend the recent work of Tuminaro, Walker and Shadid [2002] on detecting the stagnation of Newton method using the angle between the Newton direction and the steepest descent direction. We also study a nonlinear additive Schwarz preconditioned inexact Newton method, and show that it is numerically more robust. Our discussion will be based on parallel numerical experiments on solving some high Reynolds numbers steady-state incompressible Navier-Stokes equations in the velocity-pressure formulation.

The work was partially supported by the Department of Energy, DE-FC02-01ER25479, and by the National Science Foundation, CCR-0219190, ACI-0072089 and ACI-0305666.

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Author information

Authors and Affiliations

  1. Department of Applied Mathematics, University of Colorado at Boulder, Boulder, CO, 80309, USA

    Feng-Nan Hwang

  2. Department of Computer Science, University of Colorado at Boulder, Boulder, CO, 80309, USA

    Xiao-Chuan Cai

Authors
  1. Feng-Nan Hwang
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  2. Xiao-Chuan Cai
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Editor information

Editors and Affiliations

  1. Moffett Field, CA

    Timothy J. Barth

  2. Bonn

    Michael Griebel

  3. New York

    David E. Keyes  & Tamar Schlick  & 

  4. Espoo

    Risto M. Nieminen

  5. Leuven

    Dirk Roose

  6. Fachbereich Mathematik und Informatik, Freie Universität Berlin, Arnimallee 2-6, 14195, Berlin, Germany

    Ralf Kornhuber

  7. Department of Mathematics, University of Houston, Philip G. Hoffman Hall 651, 77204-3008, Houston, TX, USA

    Ronald Hoppe

  8. Dassault Aviation, 78 Quai Marcel Dassault Cedex 300, 92552, St. Cloud, France

    Jacques Périaux

  9. Laboratoire Jacques-Louis Lions, Université Paris VI, 175 Rue de Chevaleret, 75013, Paris, France

    Olivier Pironneau

  10. Courant Institute of Mathematical Science, New York University, Mercer Street 251, 10012, New York, USA

    Olof Widlund

  11. Department of Mathematics Eberly College of Science, Pennsylvania Sate University, McAllister Building 218, 16802, University Park, PA, USA

    Jinchao Xu

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© 2005 Springer-Verlag Berlin Heidelberg

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Hwang, FN., Cai, XC. (2005). Improving Robustness and Parallel Scalability of Newton Method Through Nonlinear Preconditioning. In: Barth, T.J., et al. Domain Decomposition Methods in Science and Engineering. Lecture Notes in Computational Science and Engineering, vol 40. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-26825-1_17

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