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

Abstract

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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Funding

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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Correspondence to A. I. Zadorin or N. A. Zadorin.

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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). https://doi.org/10.1134/S0965542522030149

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  • DOI: https://doi.org/10.1134/S0965542522030149

Keywords:

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