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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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A second-order overlapping Schwarz method for a 2D singularly perturbed semilinear reaction-diffusion problem
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by Natalia Kopteva and Maria Pickett PDF
Math. Comp. 81 (2012), 81-105 Request permission


An overlapping Schwarz domain decomposition is applied to a semilinear reaction-diffusion equation posed in a smooth two-dimensional domain. The problem may exhibit multiple solutions; its diffusion parameter $\varepsilon ^2$ is arbitrarily small, which induces boundary layers. The Schwarz method invokes a boundary-layer subdomain and an interior subdomain, the narrow subdomain overlap being of width $O(\varepsilon |\ln h|)$, where $h$ is the maximum side length of mesh elements, and the global number of mesh nodes does not exceed $O(h^{-2})$. We employ finite differences on layer-adapted meshes of Bakhvalov and Shishkin types in the boundary-layer subdomain, and lumped-mass linear finite elements on a quasiuniform Delaunay triangulation in the interior subdomain. For this iterative method, we present maximum norm error estimates for $\varepsilon \in (0,1]$. It is shown, in particular, that when $\varepsilon \le C|\ln h|^{-1}$, one iteration is sufficient to get second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the maximum norm uniformly in $\varepsilon$. Numerical results are presented to support our theoretical conclusions.
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Additional Information
  • Natalia Kopteva
  • Affiliation: Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland
  • MR Author ID: 610720
  • ORCID: 0000-0001-7477-6926
  • Email:
  • Maria Pickett
  • Affiliation: Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland
  • Address at time of publication: Department of Mathematics, Lion Gate Building, Lion Terrace, Portsmouth, Hampshire PO1 3HF, United Kingdom
  • Email:
  • Received by editor(s): December 11, 2009
  • Received by editor(s) in revised form: November 4, 2010
  • Published electronically: July 18, 2011
  • Additional Notes: This research was supported by an Irish Research Council for Science and Technology (IRCSET) postdoctoral fellowship and a Science Foundation Ireland grant under the Research Frontiers Programme 2008.
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 81 (2012), 81-105
  • MSC (2010): Primary 65N06, 65N15, 65N30, 65N50, 65N55
  • DOI:
  • MathSciNet review: 2833488