Nonlinear preconditioning : how to use a nonlinear Schwarz method to precondition Newton's method

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Dolean, V. and Gander, M. J. and Kheriji, W. and Kwok, F. and Masson, R. (2016) Nonlinear preconditioning : how to use a nonlinear Schwarz method to precondition Newton's method. SIAM Journal on Scientific Computing, 38 (6). A3357-A3380. ISSN 1064-8275 (https://doi.org/10.1137/15M102887X)

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Abstract

For linear problems, domain decomposition methods can be used directly as iterative solvers, but also as preconditioners for Krylov methods. In practice, Krylov acceleration is almost always used, since the Krylov method finds a much better residual polynomial than the stationary iteration, and thus converges much faster. We show in this paper that also for non-linear problems, domain decomposition methods can either be used directly as iterative solvers, or one can use them as preconditioners for Newton's method. For the concrete case of the parallel Schwarz method, we show that we obtain a preconditioner we call RASPEN (Restricted Additive Schwarz Preconditioned Exact Newton) which is similar to ASPIN (Additive Schwarz Preconditioned Inexact Newton), but with all components directly defined by the iterative method. This has the advantage that RASPEN already converges when used as an iterative solver, in contrast to ASPIN, and we thus get a substantially better preconditioner for Newton's method. The iterative construction also allows us to naturally define a coarse correction using the multigrid full approximation scheme, which leads to a convergent two level non-linear iterative domain decomposition method and a two level RASPEN non-linear preconditioner. We illustrate our findings with numerical results on the Forchheimer equation and a non-linear diffusion problem.

ORCID iDs

Dolean, V. ORCID logoORCID: https://orcid.org/0000-0002-5885-1903, Gander, M. J., Kheriji, W., Kwok, F. and Masson, R.;
  • Item type:Article
    ID code:59253
    Dates:
    Date
    Event
    1 November 2016
    Published
    22 July 2016
    Accepted
    Notes:First Published in SIAM Journal on Scientific Computing in 2016, published by the Society for Industrial and Applied Mathematics (SIAM). Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
    Subjects:Science > Mathematics
    Department:Faculty of Science > Mathematics and Statistics
    Depositing user: Pure Administrator
    Date deposited:04 Jan 2017 12:00
    Last modified:16 Dec 2024 21:54
    URI:https://strathprints.strath.ac.uk/id/eprint/59253
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