Skip to main content

On the Parameter-Uniform Convergence of Exponential Spline Interpolation in the Presence of a Boundary Layer


The paper is concerned with the problem of generalized spline interpolation of functions having large-gradient regions. Splines of the class C2, represented on each interval of the grid by the sum of a second-degree polynomial and a boundary layer function, are considered. The existence and uniqueness of the interpolation L-spline are proven, and asymptotically exact two-sided error estimates for the class of functions with an exponential boundary layer are obtained. It is established that the cubic and parabolic interpolation splines are limiting for the solution of the given problem. The results of numerical experiments are presented.

This is a preview of subscription content, access via your institution.


  1. N. S. Bakhvalov, “The optimization of methods of solving boundary value problems with a boundary layer,” USSR Comput. Math. Math. Phys. 9 (4), 139–166 (1969).

    MathSciNet  Article  MATH  Google Scholar 

  2. G. I. Shishkin, Grid Approximations of Singularly Perturbed Elliptic and Parabolic Equations (Ural. Otd. Ross. Akad. Nauk, Yekaterinburg, 1992) [in Russian].

    Google Scholar 

  3. A. M. Il’in, “Differencing scheme for a differential equation with a small parameter affecting the highest derivative,” Math. Notes 6 (2), 596–602 (1969).

    Article  MATH  Google Scholar 

  4. E. Doolan, J. Miller, and W. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers (Boole, Dublin, 1980).

    MATH  Google Scholar 

  5. I. A. Blatov, A. I. Zadorin, and E. V. Kitaeva, “Cubic spline interpolation of functions with high gradients in boundary layers,” Comput. Math. Math. Phys. 57 (1), 7–25 (2017).

    MathSciNet  Article  MATH  Google Scholar 

  6. A. I. Zadorin and M. V. Guryanova, “Analogue of a cubic spline for a function with a boundary layer component,” Proceedings of the Fifth Conference on Finite Difference Methods: Theory and Applications (Rousse Univ, Rousse, 2011), pp. 166–173.

    Google Scholar 

  7. A. I. Zadorin, “Spline interpolation of functions with a boundary layer component,” Int. J. Numer Anal. Model., Ser. B 2 (2–3), 262–279 (2011).

    MathSciNet  MATH  Google Scholar 

  8. Yu. S. Volkov, “Interpolation by splines of even degree according to Subbotin and Marsden,” Ukr. Math. J. 66 (7), 994–1012 (2014).

    MathSciNet  Article  MATH  Google Scholar 

  9. J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, The Theory of Splines and Their Applications (Academic, New York, 1967).

    MATH  Google Scholar 

  10. E. V. Strelkova and V. T. Shevaldin, “On uniform Lebesgue constants of local exponential splines with equidistant knots,” Proc. Steklov Inst. Math. 296, Suppl. 1, 206–217 (2015).

    MathSciNet  MATH  Google Scholar 

  11. Yu. S. Zav’yalov, B. N. Kvasov, and V. L. Miroshnichenko, Methods of Spline Functions (Nauka, Moscow, 1980) [in Russian].

    MATH  Google Scholar 

  12. S. B. Stechkin and Yu. N. Subbotin, Splines in Computational Mathematics (Nauka, Moscow, 1976) [in Russian].

    MATH  Google Scholar 

  13. C. de Boor, Practical Guide to Splines (Springer-Verlag, New York, 1978).

    Book  MATH  Google Scholar 

  14. G. M. Fikhtengol’ts, A Course in Differential and Integral Calculus (Nauka, Moscow, 1970), Vol. 2 [in Russian].

  15. Yu. S. Volkov, “On finding a complete interpolation spline via B-splines,” Sib. Elektron. Mat. Izv. 5, 334–338 (2008).

    MATH  Google Scholar 

  16. I. A. Blatov and E. V. Kitaeva, “Convergence of a Bakhvalov grid adaptation method for singularly perturbed boundary value problems,” Numer. Anal. Appl. 9 (1), 34–44 (2016).

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to I. A. Blatov.

Additional information

Original Russian Text © I.A. Blatov, A.I. Zadorin, E.V. Kitaeva, 2018, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2018, Vol. 58, No. 3, pp. 365–382.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Blatov, I.A., Zadorin, A.I. & Kitaeva, E.V. On the Parameter-Uniform Convergence of Exponential Spline Interpolation in the Presence of a Boundary Layer. Comput. Math. and Math. Phys. 58, 348–363 (2018).

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI:


  • singular perturbation
  • boundary layer
  • exponential spline
  • error estimate
  • uniform convergence