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Boundary layer preconditioners for finite-element discretizations of singularly perturbed reaction-diffusion problems

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Abstract

We consider the iterative solution of linear systems of equations arising from the discretization of singularly perturbed reaction-diffusion differential equations by finite-element methods on boundary-fitted meshes. The equations feature a perturbation parameter, which may be arbitrarily small, and correspondingly, their solutions feature layers: regions where the solution changes rapidly. Therefore, numerical solutions are computed on specially designed, highly anisotropic layer-adapted meshes. Usually, the resulting linear systems are ill-conditioned, and so, careful design of suitable preconditioners is necessary in order to solve them in a way that is robust, with respect to the perturbation parameter, and efficient. We propose a boundary layer preconditioner, in the style of that introduced by MacLachlan and Madden for a finite-difference method (MacLachlan and Madden, SIAM J. Sci. Comput. 35(5), A2225–A2254 2013). We prove the optimality of this preconditioner and establish a suitable stopping criterion for one-dimensional problems. Numerical results are presented which demonstrate that the ideas extend to problems in two dimensions.

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Authors and Affiliations

  1. Department of Mathematics, Ohlone College, 43600 Mission Blvd., Fremont, CA, 94539, USA

    Thái Anh Nhan

  2. Department of Mathematics and Statistics, Memorial University of Newfoundland, St John’s, NL, Canada

    Scott MacLachlan

  3. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland Galway, Galway, Ireland

    Niall Madden

Authors
  1. Thái Anh Nhan
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  2. Scott MacLachlan
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  3. Niall Madden
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Corresponding author

Correspondence to Niall Madden.

Additional information

The work of SM was partially supported by an NSERC Discovery Grant; the work of TAN was supported by the Irish Research Council under Grant No. RS/2011/179.

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Nhan, T.A., MacLachlan, S. & Madden, N. Boundary layer preconditioners for finite-element discretizations of singularly perturbed reaction-diffusion problems. Numer Algor 79, 281–310 (2018). https://doi.org/10.1007/s11075-017-0437-3

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  • DOI: https://doi.org/10.1007/s11075-017-0437-3

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