Analogue of Cubic Spline for Functions with Large Gradients in a Boundary Layer

  • Alexander ZadorinEmail author
  • Igor’ Blatov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)


The problem of spline interpolation of functions with large gradients in the boundary layer is studied. It is assumed that the function contains the known up to a factor boundary layer component responsible for the large gradients of this function in the boundary layer. A modification of the cubic spline, based on the fitting to the boundary layer component is proposed. The questions of existence, uniqueness and accuracy of such spline are investigated. Estimates of the interpolation error which are uniform with respect to a small parameter are obtained.


Function and boundary layer Uniform grid Generalized spline Interpolation Error estimation 



Supported by the program of fundamental scientific researches of the SB RAS 1.1.3., project 0314-2019-0009.


  1. 1.
    Il’in, A.M.: Differencing scheme for a differential equation with a small parameter affecting the highest derivative. USSR Math. Notes 6, 596–602 (1969)CrossRefGoogle Scholar
  2. 2.
    Bakhvalov, N.S.: The optimization of methods of solving boundary value problems with a boundary layer. USSR Comput. Math. Math. Phys. 9, 139–166 (1969)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Shishkin, G.I.: Grid Approximations of Singular Perturbation Elliptic and Parabolic Equations. UB RAS, Yekaterinburg (1992). (in Russian)zbMATHGoogle Scholar
  4. 4.
    Blatov, I.A., Zadorin, A.I., Kitaeva, E.V.: Parabolic spline interpolation for functions with large gradient in the boundary layer. Siber. Math. J. 58(4), 578–590 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Blatov, I.A., Zadorin, A.I., Kitaeva, E.V.: On the uniform convergence of parabolic spline interpolation on the class of functions with large gradients in the boundary layer. Numer. Anal. Appl. 10(2), 108–119 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Blatov, I.A., Zadorin, A.I., Kitaeva, E.V.: Cubic Spline Interpolation of Functions with High Gradients in Boundary Layers. Comput. Math. Math. Phys. 57(1), 9–28 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Zadorin, A.I.: Spline interpolation of functions with a boundary layer component. Int. J. Numer. Anal. Model Series B 2(2–3), 262–279 (2011)MathSciNetzbMATHGoogle Scholar
  8. 8.
    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. Math. Phys. 58(3), 348–363 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kellogg, R.B., Tsan, A.: Analysis of some difference approximations for a singular perturbation problem without turning points. Math. Comput. 32, 1025–1039 (1978)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Sobolev Institute of Mathematics, Siberian Branch RASNovosibirskRussia
  2. 2.Povolzhskiy State University of Telecommunications and InformaticsSamaraRussia

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