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Numerical Study of Maximum Norm a Posteriori Error Estimates for Singularly Perturbed Parabolic Problems

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Book cover Numerical Analysis and Its Applications (NAA 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8236))

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Abstract

A second-order singularly perturbed parabolic equation in one space dimension is considered. For this equation, we give computable a posteriori error estimates in the maximum norm for two semidiscretisations in time and a full discretisation using P 1 FEM in space. Both the Backward-Euler method and the Crank-Nicolson method are considered. Certain critical details of the implementation are addressed. Based on numerical results we discuss various aspects of the error estimators in particular their effectiveness.

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References

  1. Bakhvalov, N.S.: Towards optimization of methods for solving boundary value problems in the presence of boundary layers. Zh. Vychisl. Mat. i Mat. Fiz. 9, 841–859 (1969)

    MATH  Google Scholar 

  2. Friedman, A.: Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs (1964)

    MATH  Google Scholar 

  3. Kopteva, N.: Maximum norm a posteriori error estimates for a 1D singularly perturbed semilinear reaction-diffusion problem. IMA J. Numer. Anal. 27, 576–592 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Kopteva, N., Linß, T.: A posteriori error estimation for parabolic problems using elliptic reconstructions. I: Backward-Euler and Crank-Nicolson methods. Univerisity of Limerick (2011) (preprint)

    Google Scholar 

  5. Kopteva, N., Linß, T.: Maximum norm a posteriori error estimation for a time-dependent reaction-diffusion problem. Comp. Meth. Appl. Math. 12(2), 189–205 (2012)

    Google Scholar 

  6. Linß, T.: Maximum-norm error analysis of a non-monotone FEM for a singularly perturbed reaction-diffusion problem. BIT Numer. Math. 47, 379–391 (2007)

    Article  MATH  Google Scholar 

  7. Linß, T.: Layer-adapted meshes for reaction-convection-diffusion problems. Lecture Notes in Mathematics, vol. 1985. Springer, Berlin (2010)

    Google Scholar 

  8. Makridakis, C., Nochetto, R.H.: Elliptic reconstruction and a posteriori error estimates for parabolic problems. SIAM J. Numer. Anal. 41, 1585–1594 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Roos, H.-G., Stynes, M., Tobiska, L.: Robust numerical methods for singularly perturbed differential equations. Springer Series in Computational Mathematics, vol. 24. Springer, Berlin (2008)

    MATH  Google Scholar 

  10. Schmidt, A., Siebert, K.G.: Design of adaptive finite element software. Lecture Notes in Computational Science and Engineering, vol. 42. Springer, Berlin (1976)

    Google Scholar 

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Author information

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, Universitätsstr. 11, 58095, Hagen, Germany

    Torsten Linß

Authors
  1. Natalia Kopteva
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  2. Torsten Linß
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Editor information

Editors and Affiliations

  1. Institute of Information and Communication Technlogies, Bulgarian Academy of Sciences, Acad. G. Bonchev, Bl25A, 1113, Sofia, Bulgaria

    Ivan Dimov

  2. Department of Applied Analysis, Pázm\’{a}ny P\’{e}ter s\’{e}t\’{a}ny 1/C,, Eötvös Loránd University, 1117, Budapest, Hungary

    István Faragó

  3. University of Rousse, 8 Studentska Str.,, 7017, Rousse, Bulgaria

    Lubin Vulkov

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Kopteva, N., Linß, T. (2013). Numerical Study of Maximum Norm a Posteriori Error Estimates for Singularly Perturbed Parabolic Problems. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Numerical Analysis and Its Applications. NAA 2012. Lecture Notes in Computer Science, vol 8236. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41515-9_5

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  • DOI: https://doi.org/10.1007/978-3-642-41515-9_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-41514-2

  • Online ISBN: 978-3-642-41515-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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