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Energy-Norm A Posteriori Error Estimates for Singularly Perturbed Reaction-Diffusion Problems on Anisotropic Meshes: Neumann Boundary Conditions

  • Natalia Kopteva
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 120)

Abstract

Residual-type a posteriori error estimates in the energy norm are given for singularly perturbed semilinear reaction-diffusion equations posed in polygonal domains. Linear finite elements are considered on anisotropic triangulations. The error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter. The case of the Dirichlet boundary conditions was considered in the recent article (Kopteva, Numer. Math., 2017, Published online 2 May 2017. doi:10.1007/s00211-017-0889-3). Now we extend this analysis to also allow boundary conditions of Neumann type.

References

  1. 1.
    Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley-Interscience, New York (2000)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bakhvalov, N.S.: On the optimization of methods for solving boundary value problems with boundary layers. Zh. Vychisl. Mat. Mat. Fis. 9, 841–859 (1969) (in Russian)MathSciNetGoogle Scholar
  3. 3.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Demlow, A., Kopteva, N.: Maximum-norm a posteriori error estimates for singularly perturbed elliptic reaction-diffusion problems. Numer. Math. 133, 707–742 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Kopteva, N.: Maximum norm error analysis of a 2d singularly perturbed semilinear reaction-diffusion problem. Math. Comput. 76, 631–646 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Kopteva, N.: Maximum norm a posteriori error estimate for a 2d singularly perturbed reaction-diffusion problem. SIAM J. Numer. Anal. 46, 1602–1618 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Kopteva, N.: Linear finite elements may be only first-order pointwise accurate on anisotropic triangulations. Math. Comput. 83, 2061–2070 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Kopteva, N.: Maximum-norm a posteriori error estimates for singularly perturbed reaction-diffusion problems on anisotropic meshes. SIAM J. Numer. Anal. 53, 2519–2544 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Kopteva, N.: Energy-norm a posteriori error estimates for singularly perturbed reaction-diffusion problems on anisotropic meshes. Numer. Math. (2017). Published online 2 May 2017. doi:10.1007/s00211-017-0889-3Google Scholar
  10. 10.
    Kopteva, N.: Fully computable a posteriori error estimator using anisotropic flux equilibration on anisotropic meshes. (2017, submitted for publication). http://www.staff.ul.ie/natalia/pubs.html
  11. 11.
    Kopteva, N., O’Riordan, E.: Shishkin meshes in the numerical solution of singularly perturbed differential equations. Int. J. Numer. Anal. Model. 7, 393–415 (2010)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Kunert, G.: An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes. Numer. Math. 86, 471–490 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Kunert, G.: Robust a posteriori error estimation for a singularly perturbed reaction-diffusion equation on anisotropic tetrahedral meshes. Adv. Comput. Math. 15, 237–259 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Kunert, G., Verfürth, R.: Edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes. Numer. Math. 86, 283–303 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Nochetto, R.H.: Pointwise a posteriori error estimates for monotone semi-linear equations. Lecture Notes at 2006 CNA Summer School Probabilistic and Analytical Perspectives on Contemporary PDEs (2006). http://www.math.cmu.edu/cna/Summer06/lecturenotes/nochetto/
  16. 16.
    Roos, H.-G., Stynes, M., Tobiska, T.: Robust Numerical Methods for Singularly Perturbed Differential Equations. Springer, Berlin (2008)zbMATHGoogle Scholar
  17. 17.
    Siebert, K.G.: An a posteriori error estimator for anisotropic refinement. Numer. Math. 73, 373–398 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Verfürth, R.: Robust a posteriori error estimators for a singularly perturbed reaction-diffusion equation. Numer. Math. 78, 479–493 (1998)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of LimerickLimerickIreland

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