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Cubic spline interpolation of functions with high gradients in boundary layers


The cubic spline interpolation of grid functions with high-gradient regions is considered. Uniform meshes are proved to be inefficient for this purpose. In the case of widely applied piecewise uniform Shishkin meshes, asymptotically sharp two-sided error estimates are obtained in the class of functions with an exponential boundary layer. It is proved that the error estimates of traditional spline interpolation are not uniform with respect to a small parameter, and the error can increase indefinitely as the small parameter tends to zero, while the number of nodes N is fixed. A modified cubic interpolation spline is proposed, for which O((ln N/N)4) error estimates that are uniform with respect to the small parameter are obtained.

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  1. 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 

  2. 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 

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

    Google Scholar 

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

    MATH  Google Scholar 

  5. Yu. S. Zav’yalov, B. N. Kvasov, and V. L. Miroshnichenko, Methods of Spline Functions (Nauka, Moscow, 1981) [in Russian].

    MATH  Google Scholar 

  6. A. I. Zadorin, “Method of interpolation for a boundary layer problem,” Sib. Zh. Vychisl. Mat. 10 (3), 267–275 (2007).

    MATH  Google Scholar 

  7. 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 

  8. A. I. Zadorin, “Lagrange interpolation and Newton–Cotes formulas for functions with a boundary layer component on piecewise uniform meshes,” Numer. Anal. Appl. 8 (3), 235–247 (2015).

    MathSciNet  Article  MATH  Google Scholar 

  9. N. L. Zmatrakov, “Convergence of an interpolation process for parabolic and cubic splines,” Proc. Steklov Inst. Math. 138, 75–99 (1977).

    MathSciNet  MATH  Google Scholar 

  10. N. L. Zmatrakov, “A necessary condition for convergence of interpolating parabolic and cubic splines,” Math. Notes 19 (2), 100–107 (1976).

    MathSciNet  Article  MATH  Google Scholar 

  11. J. J. H. Miller, E. O’Riordan, and G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions (revised ed.) (World Scientific, Singapore, 2012).

    Book  MATH  Google Scholar 

  12. T. Linss, “The necessity of Shishkin decompositions,” Appl. Math. Lett. 14, 891–896 (2001).

    MathSciNet  Article  MATH  Google Scholar 

  13. C. de Boor, Practical Guide to Splines (Springer-Verlag, New York, 1978; Radio i Svyaz’, Moscow, 1985).

    MATH  Google Scholar 

  14. S. Demko, “Inverses of band matrices and local convergence of spline projections,” SIAM J. Numer. Anal. 14 (4), 616–619 (1977).

    MathSciNet  Article  MATH  Google Scholar 

  15. V. V. Voevodin and Yu. A. Kuznetsov, Matrices and Computations (Nauka, Moscow, 1984) [in Russian].

    MATH  Google Scholar 

  16. A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations (Nauka, Moscow, 1978; Birkhäuser, Basel, 1989).

    Book  Google Scholar 

  17. I. A. Blatov, “Incomplete factorization methods for systems with sparse matrices,” Comput. Math. Math. Phys. 33 (6), 727–741 (1993).

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  19. 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 

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Correspondence to I. A. Blatov or A. I. Zadorin.

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Original Russian Text © I.A. Blatov, A.I. Zadorin, E.V. Kitaeva, 2017, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2017, Vol. 57, No. 1, pp. 9–28.

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

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  • singular perturbation
  • boundary layer
  • Shishkin mesh
  • cubic spline
  • modification
  • error estimate