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Improved maximum-norm a posteriori error estimates for linear and semilinear parabolic equations

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Abstract

Linear and semilinear second-order parabolic equations are considered. For these equations, we give a posteriori error estimates in the maximum norm that improve upon recent results in the literature. In particular it is shown that logarithmic dependence on the time step size can be eliminated. Semidiscrete and fully discrete versions of the backward Euler and of the Crank-Nicolson methods are considered. For their full discretizations, we use elliptic reconstructions that are, respectively, piecewise-constant and piecewise-linear in time. Certain bounds for the Green’s function of the parabolic operator are also employed.

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Authors and Affiliations

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

    Natalia Kopteva

  2. Fakultät für Mathematik und Informatik, FernUniversität in Hagen, Universitätsstraße 11, 58095, Hagen, Germany

    Torsten Linß

Authors
  1. Natalia Kopteva
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  2. Torsten Linß
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Correspondence to Torsten Linß.

Additional information

Communicated by: Karsten Urban

Dedicated to Prof. Hans-Görg Roos on the occasion of his 65th birthday

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Kopteva, N., Linß, T. Improved maximum-norm a posteriori error estimates for linear and semilinear parabolic equations. Adv Comput Math 43, 999–1022 (2017). https://doi.org/10.1007/s10444-017-9514-3

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  • DOI: https://doi.org/10.1007/s10444-017-9514-3

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