### Abstract

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 ε2 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(ε| 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 ε ∈ (0, 1]. It is shown, in particular, that when ε ≤ 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 ε. Numerical results are presented to support our theoretical conclusions.

For this iterative method, we present maximum norm error estimates for ε ∈ (0, 1]. It is shown, in particular, that when ε ≤ 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 ε. Numerical results are presented to support our theoretical conclusions.

Original language | English |
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Pages (from-to) | 81-105 |

Number of pages | 25 |

Journal | Mathematics of Computation |

Volume | 81 |

Issue number | 277 |

Early online date | 18 Jul 2011 |

DOIs | |

Publication status | Published - Jan 2012 |

### Keywords

- semilinear reaction-diffusion
- singular perturbation
- domain decomposition
- overlapping Schwarz
- Bakhvalov mesh
- Shishkin mesh
- supra-convergence
- lumped-mass finite elements

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## Cite this

Kopteva, N., & Pickett, M. (2012). A second-order overlapping Schwarz method for a 2D singularly perturbed semilinear reaction-diffusion problem.

*Mathematics of Computation*,*81*(277), 81-105. https://doi.org/10.1090/S0025-5718-2011-02521-4