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Interpolation of a function of two variables with large gradients in boundary layers

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Abstract

The paper is concerned with the interpolation of a function of two variables with large gradients in the boundary layers. An underlying function is assumed to be the sum of the regular componentwith derivatives bounded up to some order and two boundary layer components, the latter are known up to a multiplicative factor. Such a representation is typical for the solution of a singular perturbed elliptic problem. A two-dimensional interpolation formula exact on the boundary layer components is put forward. The formula has an arbitrary number of nodes in each direction. An error estimate is obtained which is uniform on the gradients of the underlying function in boundary layers. Results of numerical experiments are provided.

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References

  1. A. I. Zadorin, Sib. Zh. Vychisl.Mat. 10 (3), 267–275 (2007).

    MathSciNet  Google Scholar 

  2. A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems (Nauka, Moscow, 1989) [in Russian].

    MATH  Google Scholar 

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

    Google Scholar 

  4. 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 (World Sci., Singapore 2012).

    Book  MATH  Google Scholar 

  5. A. M. Il’in, Math. Notes 6 (2), 596–602 (1969).

    Article  Google Scholar 

  6. A. I. Zadorin and N. A. Zadorin, Comput. Math.Math. Phys. 50 (2), 211–223 (2010).

    Article  MathSciNet  Google Scholar 

  7. A. I. Zadorin, Int. J. Numer. Anal.Model. Ser. B. 2 (2–3), 262–279 (2011).

    MathSciNet  Google Scholar 

  8. A. I. Zadorin and N. A. Zadorin, Sib. Electron.Math. Rep. 9, 445–455 (2012).

    MathSciNet  Google Scholar 

  9. N. S. Bakhvalov, N. P. Zhidkov, and G. M. Kobel’kov, Numerical Methods (Nauka, Moscow, 1987) [in Russian].

    MATH  Google Scholar 

  10. I. A. Blatov and N. A. Zadorin, Nauka Mir 2 (18), 13–17 (2015).

    Google Scholar 

  11. A. A. Kornev and E. V. Chizhonkov, Exercises in Numerical Methods (Mosk. Gos. Univ., Moscow, 2003), Part 2 [in Russian].

    Google Scholar 

  12. H. G. Roos, M. Stynes, and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Convection-Diffusion and Flow problems (Springer, Berlin, 2008).

    MATH  Google Scholar 

  13. L. G. Vulkov and A. I. Zadorin, AIP Conf. Proc. 1186 (1), 371–379 (2009).

    Article  MathSciNet  Google Scholar 

  14. A. I. Zadorin and N. A. Zadorin, Sib. Elektron.Mat. Izv. 8, 247–267 (2011).

    MathSciNet  Google Scholar 

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Correspondence to A. I. Zadorin.

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Original Russian Text © A.I. Zadorin, 2015, published in Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2015, Vol. 157, No. 2, pp. 55–67.

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Zadorin, A.I. Interpolation of a function of two variables with large gradients in boundary layers. Lobachevskii J Math 37, 349–359 (2016). https://doi.org/10.1134/S1995080216030069

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  • DOI: https://doi.org/10.1134/S1995080216030069

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