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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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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