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

  • I. A. Blatov
  • A. I. Zadorin
  • E. V. Kitaeva


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.


singular perturbation boundary layer exponential spline error estimate uniform convergence 


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  1. 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).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 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. 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).CrossRefzbMATHGoogle Scholar
  4. 4.
    E. Doolan, J. Miller, and W. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers (Boole, Dublin, 1980).zbMATHGoogle Scholar
  5. 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).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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. 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).MathSciNetzbMATHGoogle Scholar
  8. 8.
    Yu. S. Volkov, “Interpolation by splines of even degree according to Subbotin and Marsden,” Ukr. Math. J. 66 (7), 994–1012 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, The Theory of Splines and Their Applications (Academic, New York, 1967).zbMATHGoogle Scholar
  10. 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).MathSciNetzbMATHGoogle Scholar
  11. 11.
    Yu. S. Zav’yalov, B. N. Kvasov, and V. L. Miroshnichenko, Methods of Spline Functions (Nauka, Moscow, 1980) [in Russian].zbMATHGoogle Scholar
  12. 12.
    S. B. Stechkin and Yu. N. Subbotin, Splines in Computational Mathematics (Nauka, Moscow, 1976) [in Russian].zbMATHGoogle Scholar
  13. 13.
    C. de Boor, Practical Guide to Splines (Springer-Verlag, New York, 1978).CrossRefzbMATHGoogle Scholar
  14. 14.
    G. M. Fikhtengol’ts, A Course in Differential and Integral Calculus (Nauka, Moscow, 1970), Vol. 2 [in Russian].Google Scholar
  15. 15.
    Yu. S. Volkov, “On finding a complete interpolation spline via B-splines,” Sib. Elektron. Mat. Izv. 5, 334–338 (2008).zbMATHGoogle Scholar
  16. 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).MathSciNetCrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Povolzhskiy State University of Telecommunications and InformaticsSamaraRussia
  2. 2.Sobolev Institute of Mathematics, Omsk Branch, Siberian BranchRussian Academy of SciencesOmskRussia
  3. 3.Samara State Aerospace UniversitySamaraRussia

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