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Lagrange Interpolation and the Newton–Cotes Formulas on a Bakhvalov Mesh in the Presence of a Boundary Layer


Application of a Lagrange polynomial on a Bakhvalov mesh for the interpolation of a function with large gradients in an exponential boundary layer is studied. The problem is that the use of a Lagrange polynomial on a uniform mesh for interpolation of such a function can lead to errors of order \(O(1),\) despite the smallness of the mesh size. The Bakhvalov mesh is widely used for the numerical solution of singularly perturbed problems, and the analysis of interpolation formulas on such a mesh is of interest. Estimates of the error of interpolation by a Lagrange polynomial with an arbitrary number of interpolation nodes on a Bakhvalov mesh are obtained. The result is used to estimate the error of the Newton–Cotes formulas on a Bakhvalov mesh. The results of numerical experiments are presented.

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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  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  Google Scholar 

  3. T. Linß, Layer-Adapted Meshes for Reaction–Convection–Diffusion Problems (Springer-Verlag, Berlin, 2010).

    Book  Google Scholar 

  4. R. Vulanovic, “A priori meshes for singularly perturbed quasilinear two-point boundary value problems,” IMA J. Numer. Anal. 21, 349–366 (2001).

    MathSciNet  Article  Google Scholar 

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

    MATH  Google Scholar 

  6. 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 Scientific, Singapore, 2012).

    Book  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  9. 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  Google Scholar 

  10. I. A. Blatov and N. A. Zadorin, “Interpolation on Bakhvalov meshes in the case of an exponential boundary layer,” Uch. Zap. Kazan. Univ. Fiz.-Mat. Nauki 161 (4), 497–508 (2019).

    Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  12. H. G. Roos, “Layer-adapted meshes: Milestones in 50 years of history,” Appl. Math. (2019). arXiv:1909.08273v1.

  13. A. I. Zadorin and N. A. Zadorin, “Quadrature formulas for functions with a boundary-layer component,” Comput. Math. Math. Phys. 51 (11), 1837–1846 (2011).

    MathSciNet  Article  Google Scholar 

  14. A. I. Zadorin and N. A. Zadorin, “An analogue of the four-point Newton–Cotes formula for a function with a boundary-layer component,” Numer. Anal. Appl. 6 (4), 268–278 (2013).

    Article  Google Scholar 

  15. A. I. Zadorin and N. A. Zadorin, “Non-polynomial interpolation of functions with large gradients and its application,” Comput. Math. Math. Phys. 61 (2), 167–176 (2021).

    MathSciNet  Article  Google Scholar 

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The work of A.I. Zadorin was supported by the Russian Foundation for Basic Research (project no. 20-01-00650) and the Siberain Branch of the Russian Academy of Sciences (program SB RAS 1.1.3, project no. 0314-2019-0009). The work of N.A. Zadorin was supported by the Russian Foundation for Basic Research (project no. 19-31-60009).

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Translated by E. Chernokozhin

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Zadorin, A.I., Zadorin, N.A. Lagrange Interpolation and the Newton–Cotes Formulas on a Bakhvalov Mesh in the Presence of a Boundary Layer. Comput. Math. and Math. Phys. 62, 347–358 (2022).

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  • function of one variable
  • boundary layer
  • Bakhvalov mesh
  • Lagrange interpolation polynomial
  • error estimate