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Maximum norm a posteriori error estimate for a 3d singularly perturbed semilinear reaction-diffusion problem

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Abstract

A singularly perturbed semilinear reaction-diffusion problem in the unit cube, is discretized on arbitrary nonuniform tensor-product meshes. We establish a second-order maximum norm a posteriori error estimate that holds true uniformly in the small diffusion parameter. No mesh aspect ratio condition is imposed. This result is obtained by combining (i) sharp bounds on the Green’s function of the continuous differential operator in the Sobolev W 1,1 and W 2,1 norms and (ii) a special representation of the residual in terms of an arbitrary current mesh and the current computed solution. Numerical results on a priori chosen meshes are presented that support our theoretical estimate.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland

    Naresh M. Chadha & Natalia Kopteva

Authors
  1. Naresh M. Chadha
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  2. Natalia Kopteva
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Corresponding author

Correspondence to Natalia Kopteva.

Additional information

Communicated by Martin Stynes.

This publication has emanated from research conducted with the financial support of Science Foundation Ireland under the Research Frontiers Programme 2008; Grant 08/RFP/MTH1536.

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Chadha, N.M., Kopteva, N. Maximum norm a posteriori error estimate for a 3d singularly perturbed semilinear reaction-diffusion problem. Adv Comput Math 35, 33–55 (2011). https://doi.org/10.1007/s10444-010-9163-2

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  • DOI: https://doi.org/10.1007/s10444-010-9163-2

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