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Convergence and Supercloseness in a Balanced Norm of Finite Element Methods on Bakhvalov-Type Meshes for Reaction-Diffusion Problems

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Abstract

In convergence analysis of finite element methods for singularly perturbed reaction–diffusion problems, balanced norms have been successfully introduced to replace standard energy norms so that layers can be captured. In this article, we focus on the convergence analysis in a balanced norm on Bakhvalov-type rectangular meshes. In order to achieve our goal, a novel interpolation operator, which consists of a local \(L^2\) projection operator and the Lagrange interpolation operator, is introduced for a convergence analysis of optimal order in the balanced norm. The analysis also depends on the stabilities of the \(L^2\) projection and the characteristics of Bakhvalov-type meshes. Furthermore, we obtain a supercloseness result in the balanced norm, which appears in the literature for the first time. This result depends on another novel interpolant, which consists of the local \(L^2\) projection operator, a vertices-edges-element operator and some corrections on the boundary.

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References

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

    MathSciNet  Google Scholar 

  2. Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)

    Article  MathSciNet  Google Scholar 

  3. Constantinou, P., Franz, S., Ludwig, L., Xenophontos, C.: Finite element approximation of reaction-diffusion problems using an exponentially graded mesh. Comput. Math. Appl. 76(10), 2523–2534 (2018). https://doi.org/10.1016/j.camwa.2018.08.051

    Article  MathSciNet  MATH  Google Scholar 

  4. Crouzeix, M., Thomée, V.: The stability in \(L_p\) and \(W^1_p\) of the \(L_2\)-projection onto finite element function spaces. Math. Comp. 48(178), 521–532 (1987). https://doi.org/10.2307/2007825

    Article  MathSciNet  MATH  Google Scholar 

  5. Durán, R.G., Lombardi, A.L., Prieto, M.I.: Supercloseness on graded meshes for \(Q_1\) finite element approximation of a reaction-diffusion equation. J. Comput. Appl. Math. 242, 232–247 (2013). https://doi.org/10.1016/j.cam.2012.10.004

    Article  MathSciNet  MATH  Google Scholar 

  6. Franz, S.: Singularly perturbed problems with characteristic layers: Supercloseness and postprocessing. PhD thesis, Department of Mathematics, TU Dresden (2008)

  7. Franz, S., Roos, H.G.: Error estimates in balanced norms of finite element methods for higher order reaction-diffusion problems. arXiv:1902.09829 [math.NA] (2019)

  8. Han, H., Kellogg, R.B.: Differentiability properties of solutions of the equation \(-\epsilon ^2\Delta u+ru=f(x, y)\) in a square. SIAM J. Math. Anal. 21(2), 394–408 (1990). https://doi.org/10.1137/0521022

    Article  MathSciNet  MATH  Google Scholar 

  9. Kopteva, N.: On the convergence, uniform with respect to the small parameter, of a scheme with central difference on refined grids. Zh. Vychisl. Mat. Mat. Fiz. 39(10), 1662–1678 (1999)

    MathSciNet  Google Scholar 

  10. Kopteva, N., Savescu, S.B.: Pointwise error estimates for a singularly perturbed time-dependent semilinear reaction-diffusion problem. IMA J. Numer. Anal. 31(2), 616–639 (2011). https://doi.org/10.1093/imanum/drp032

    Article  MathSciNet  MATH  Google Scholar 

  11. Li, B.: Lagrange interpolation and finite element superconvergence. Numer. Methods Partial Differ. Equ. 20(1), 33–59 (2004). https://doi.org/10.1002/num.10078

    Article  MathSciNet  MATH  Google Scholar 

  12. Lin, Q., Yan, N., Zhou, A.: A rectangle test for interpolated finite elements. In: Proc. Syst. Sci. Eng., pp. 217–229. Great Wall (H.K.) Culture Publish Co. (1991)

  13. Lin, R., Stynes, M.: A balanced finite element method for singularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal. 50(5), 2729–2743 (2012). https://doi.org/10.1137/110837784

    Article  MathSciNet  MATH  Google Scholar 

  14. Linß, T.: Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems. Lecture Notes in Mathematics, vol. 1985. Springer-Verlag, Berlin (2010)

    Book  Google Scholar 

  15. Liu, X., Stynes, M., Zhang, J.: Supercloseness of edge stabilization on Shishkin rectangular meshes for convection-diffusion problems with exponential layers. IMA J. Numer. Anal. 38(4), 2105–2122 (2018). https://doi.org/10.1093/imanum/drx055

    Article  MathSciNet  MATH  Google Scholar 

  16. Roos, H., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer Series in Computational Mathematics, 2nd edn. Springer-Verlag, Berlin (2008)

    MATH  Google Scholar 

  17. Roos, H.G.: Error estimates for linear finite elements on Bakhvalov-type meshes. Appl. Math. 51(1), 63–72 (2006). https://doi.org/10.1007/s10492-006-0005-y

    Article  MathSciNet  MATH  Google Scholar 

  18. Roos, H.G., Schopf, M.: Convergence and stability in balanced norms of finite element methods on Shishkin meshes for reaction-diffusion problems. ZAMM Z. Angew. Math. Mech. 95(6), 551–565 (2015). https://doi.org/10.1002/zamm.201300226

    Article  MathSciNet  MATH  Google Scholar 

  19. Shishkin, G.I.: Grid approximation of singularly perturbed elliptic and parabolic equations (in Russian). Second doctoral thesis, Keldysh Institute, Moscow (1990)

  20. Stynes, M.: Steady-state convection-diffusion problems. Acta Numer. 14, 445–508 (2005). https://doi.org/10.1017/S0962492904000261

    Article  MathSciNet  MATH  Google Scholar 

  21. Stynes, M., Tobiska, L.: Using rectangular \(Q_p\) elements in the SDFEM for a convection-diffusion problem with a boundary layer. Appl. Numer. Math. 58(12), 1789–1802 (2008). https://doi.org/10.1016/j.apnum.2007.11.004

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang, J., Liu, X.: Analysis of SDFEM on Shishkin triangular meshes and hybrid meshes for problems with characteristic layers. J. Sci. Comput. 68(3), 1299–1316 (2016). https://doi.org/10.1007/s10915-016-0180-2

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, J., Liu, X.: Supercloseness of the SDFEM on Shishkin triangular meshes for problems with exponential layers. Adv. Comput. Math. 43(4), 759–775 (2017). https://doi.org/10.1007/s10444-016-9505-9

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhang, J., Liu, X.: Pointwise estimates of SDFEM on Shishkin triangular meshes for problems with exponential layers. BIT 58(1), 221–246 (2018). https://doi.org/10.1007/s10543-017-0661-1

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhang, J., Liu, X.: Supercloseness of continuous interior penalty methods on Shishkin triangular meshes and hybrid meshes for singularly perturbed problems with characteristic layers. J. Sci. Comput. 76(3), 1633–1656 (2018). https://doi.org/10.1007/s10915-018-0677-y

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhang, J., Liu, X.: Convergence of a finite element method on a Bakhvalov-type mesh for singularly perturbed reaction-diffusion equation. Appl. Math. Comput. 385, 125,403 (2020). https://doi.org/10.1016/j.amc.2020.125403

    Article  MathSciNet  Google Scholar 

  27. Zhang, J., Liu, X.: Optimal order of uniform convergence for finite element method on Bakhvalov-type meshes. J. Sci. Comput. 85(1), 2 (2020). https://doi.org/10.1007/s10915-020-01312-y

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhang, J., Liu, X.: Supercloseness of linear finite element method on Bakhvalov-type meshes for singularly perturbed convection-diffusion equation in 1D. Appl. Math. Lett. 111, 106,624 (2021). https://doi.org/10.1016/j.aml.2020.106624

    Article  MathSciNet  Google Scholar 

  29. Zhang, J., Liu, X., Yang, M.: Optimal order \(L^2\) error estimate of SDFEM on Shishkin triangular meshes for singularly perturbed convection-diffusion equations. SIAM J. Numer. Anal. 54(4), 2060–2080 (2016). https://doi.org/10.1137/15M101035X

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhang, J., Stynes, M.: Supercloseness of continuous interior penalty method for convection-diffusion problems with characteristic layers. Comput. Methods Appl. Mech. Engrg. 319, 549–566 (2017). https://doi.org/10.1016/j.cma.2017.03.013

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank the two anonymous referees for their valuable comments and suggestions that led us to improve this paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, China

    Jin Zhang

  2. School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan, 250353, China

    Xiaowei Liu

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  1. Jin Zhang
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  2. Xiaowei Liu
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Correspondence to Jin Zhang.

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

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Zhang, J., Liu, X. Convergence and Supercloseness in a Balanced Norm of Finite Element Methods on Bakhvalov-Type Meshes for Reaction-Diffusion Problems. J Sci Comput 88, 27 (2021). https://doi.org/10.1007/s10915-021-01542-8

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  • DOI: https://doi.org/10.1007/s10915-021-01542-8

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