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Finite element method for singularly perturbed problems with two parameters on a Bakhvalov-type mesh in 2D

Abstract

On a Bakhvalov-type mesh widely used for boundary layers, we consider the finite element method for singularly perturbed elliptic problems with two parameters on the unit square. It is a very challenging task to analyze uniform convergence of finite element method on this mesh in 2D. The existing analysis tool, quasi-interpolation, is only applicable to one-dimensional case because of the complexity of Bakhvalov-type mesh in 2D. In this paper, a powerful tool, Lagrange-type interpolation, is proposed, which is simple and effective and can be used in both 1D and 2D. The application of this interpolation in 2D must be handled carefully. Some boundary correction terms must be introduced to maintain the homogeneous Dirichlet boundary condition. These correction terms are difficult to be handled because the traditional analysis do not work for them. To overcome this difficulty, we derive a delicate estimation of the width of some mesh. Moreover, we adopt different analysis strategies for different layers. Finally, we prove uniform convergence of optimal order. Numerical results verify the theoretical analysis.

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References

  1. 1.

    Apel, T.: Anisotropic finite elements: local estimates and applications. Advances in Numerical Mathematics. B. G, Teubner, Stuttgart (1999)

  2. 2.

    Bahvalov, N.S.: On the optimization of the methods for solving boundary value problems in the presence of a boundary layer. ž. vyčisl. Mat i Mat Fiz. 9, 841–859 (1969)

    MathSciNet  Google Scholar 

  3. 3.

    Brdar, M., Zarin, H.: A singularly perturbed problem with two parameters on a Bakhvalov-type mesh. J. Comput. Appl Math. 292, 307–319 (2016)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods, Volume 15 of Texts in Applied Mathematics, third. Springer, New York (2008)

    Book  Google Scholar 

  5. 5.

    Ciarlet, P.G.: The finite element method for elliptic problems, Volume 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM). Philadelphia, PA (2002)

  6. 6.

    Linß, T., Stynes, M.: Asymptotic analysis and Shishkin-type decomposition for an elliptic convection-diffusion problem. J. Math. Anal Appl. 261 (2), 604–632 (2001)

    MathSciNet  Article  Google Scholar 

  7. 7.

    O’Malley, Jr. R.E.: Two-parameter singular perturbation problems for second-order equations. J. Math. Mech. 16, 1143–1164 (1967)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Roos, H.-G.: Error estimates for linear finite elements on Bakhvalov-type meshes. Appl. Math. 51(1), 63–72 (2006)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Shishkin, G.I.: Grid approximation of singularly perturbed elliptic equations in case of limit zero-order equations degenerating at the boundary. Soviet. J. Numer. Anal. Math. Modelling 5(6), 523–548 (1990)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Teofanov, L.J., Roos, H.-G.: An elliptic singularly perturbed problem with two parameters. I. Solution decomposition. J. Comput. Appl Math. 206(2), 1082–1097 (2007)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Teofanov, L.J., Roos, H.-G.: An elliptic singularly perturbed problem with two parameters. II. Robust finite element solution. J. Comput. Appl Math. 212(2), 374–389 (2008)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Zhang, J., Liu, X.: Optimal order of uniform convergence for finite element method on Bakhvalov-type meshes. J. Sci Comput. 85(1), 2, 14 (2020)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank the anonymous referees for his/her valuable comments and suggestions that led us to improve this paper.

Funding

This research is supported by the National Natural Science Foundation of China (11771257, 11601251).

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Correspondence to Jin Zhang.

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Zhang, J., Lv, Y. Finite element method for singularly perturbed problems with two parameters on a Bakhvalov-type mesh in 2D. Numer Algor (2021). https://doi.org/10.1007/s11075-021-01194-7

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Keywords

  • Singular perturbation
  • Convection–diffusion equation
  • Finite element method
  • Bakhvalov-type mesh
  • Two parameters

Mathematics Subject Classification (2010)

  • 65N12
  • 65N30
  • 65N50