Volume 16, Issue 4
Formulas of Numerical Differentiation on a Uniform Mesh for Functions with the Exponential Boundary Layer

Alexander Zadorin & Svetlana Tikhovskaya

DOI:

Int. J. Numer. Anal. Mod., 16 (2019), pp. 590-608.

Published online: 2019-02

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  • Abstract

It is known that the solution of a singularly perturbed problem corresponds to the function with large gradients in a boundary layer. The application of Lagrange polynomial on a uniform mesh to interpolate such functions leads to large errors. To achieve the error estimates uniform with respect to a small parameter, we can use either a polynomial interpolation on a mesh which condenses in a boundary layer or we can use special interpolation formulas which are exact on a boundary layer component of the interpolating function. In this paper, we construct and study the formulas of numerical differentiation based on the interpolation formulas which are exact on a boundary layer component. We obtained the error estimates which are uniform with respect to a small parameter. Some numerical results validating the theoretical estimates are discussed.

  • Keywords

Function of one variable, exponential boundary layer, formulas of numerical differentiation, an error estimate.

  • AMS Subject Headings

65D25, 41A30

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

zadorin@ofim.oscsbras.ru (Alexander Zadorin)

s.tihovskaya@yandex.ru (Svetlana Tikhovskaya)

  • BibTex
  • RIS
  • TXT
@Article{IJNAM-16-590, author = {Zadorin , Alexander and Tikhovskaya , Svetlana }, title = {Formulas of Numerical Differentiation on a Uniform Mesh for Functions with the Exponential Boundary Layer}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2019}, volume = {16}, number = {4}, pages = {590--608}, abstract = {

It is known that the solution of a singularly perturbed problem corresponds to the function with large gradients in a boundary layer. The application of Lagrange polynomial on a uniform mesh to interpolate such functions leads to large errors. To achieve the error estimates uniform with respect to a small parameter, we can use either a polynomial interpolation on a mesh which condenses in a boundary layer or we can use special interpolation formulas which are exact on a boundary layer component of the interpolating function. In this paper, we construct and study the formulas of numerical differentiation based on the interpolation formulas which are exact on a boundary layer component. We obtained the error estimates which are uniform with respect to a small parameter. Some numerical results validating the theoretical estimates are discussed.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/13016.html} }
TY - JOUR T1 - Formulas of Numerical Differentiation on a Uniform Mesh for Functions with the Exponential Boundary Layer AU - Zadorin , Alexander AU - Tikhovskaya , Svetlana JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 590 EP - 608 PY - 2019 DA - 2019/02 SN - 16 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/13016.html KW - Function of one variable, exponential boundary layer, formulas of numerical differentiation, an error estimate. AB -

It is known that the solution of a singularly perturbed problem corresponds to the function with large gradients in a boundary layer. The application of Lagrange polynomial on a uniform mesh to interpolate such functions leads to large errors. To achieve the error estimates uniform with respect to a small parameter, we can use either a polynomial interpolation on a mesh which condenses in a boundary layer or we can use special interpolation formulas which are exact on a boundary layer component of the interpolating function. In this paper, we construct and study the formulas of numerical differentiation based on the interpolation formulas which are exact on a boundary layer component. We obtained the error estimates which are uniform with respect to a small parameter. Some numerical results validating the theoretical estimates are discussed.

Alexander Zadorin & Svetlana Tikhovskaya. (2019). Formulas of Numerical Differentiation on a Uniform Mesh for Functions with the Exponential Boundary Layer. International Journal of Numerical Analysis and Modeling. 16 (4). 590-608. doi:
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