Geodesic
Application of Taylor's formula to polynomial approximation of a function of two variables with large gradients
Matematičeskie trudy, Tome 27 (2024) no. 4, pp. 81-92 
A. I. Zadorin. Application of Taylor's formula to polynomial approximation of a function of two variables with large gradients. Matematičeskie trudy, Tome 27 (2024) no. 4, pp. 81-92. http://geodesic.mathdoc.fr/item/MT_2024_27_4_a4/
@article{MT_2024_27_4_a4,
     author = {A. I. Zadorin},
     title = {Application of {Taylor's} formula to polynomial approximation of a function of two variables with large gradients},
     journal = {Matemati\v{c}eskie trudy},
     pages = {81--92},
     year = {2024},
     volume = {27},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2024_27_4_a4/}
}
TY  - JOUR
AU  - A. I. Zadorin
TI  - Application of Taylor's formula to polynomial approximation of a function of two variables with large gradients
JO  - Matematičeskie trudy
PY  - 2024
SP  - 81
EP  - 92
VL  - 27
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/MT_2024_27_4_a4/
LA  - ru
ID  - MT_2024_27_4_a4
ER  - 
%0 Journal Article
%A A. I. Zadorin
%T Application of Taylor's formula to polynomial approximation of a function of two variables with large gradients
%J Matematičeskie trudy
%D 2024
%P 81-92
%V 27
%N 4
%U http://geodesic.mathdoc.fr/item/MT_2024_27_4_a4/
%G ru
%F MT_2024_27_4_a4

Voir la notice de l'article provenant de la source Math-Net.Ru

The problem of approximating a function of two variables with large gradients by polynomials based on the Taylor formula is investigated. It is assumed that the decomposition of the function in the form of a sum of regular and boundary layer components is valid. The boundary layer component is known with an accuracy of up to a factor and is responsible for large gradients of the function. Such a decomposition is valid for the solution of singularly perturbed elliptic problem. The problem is that approximating such a function by polynomials based on the Taylor formula can lead to significant errors due to the presence of the boundary layer component. A formula for approximating a function is developed, using the Taylor formula and, by construction, being exact on the boundary layer component of the given function of two variables. It is proved that the error estimate of the constructed formula depends on the partial derivatives of the regular component and does not depend on the derivatives of the boundary layer component, which significantly increases the accuracy of approximating the function by polynomials.

[1] Zorich V. A., Mathematical analysis, Springer, Berlin, 2015 | MR

[2] Shishkin G. I., Mesh approximations of singularly perturbed elliptic and parabolic equations, Ural Otd. Ross. Akad. Nauk, Yekaterinburg, 1992 (in Russian)

[3] Kellogg R. B., Tsan A., “Analysis of some difference approximations for a singular perturbation problem without turning points”, Math. Comput., 32 (1978), 1025–1039 | DOI | MR

[4] Roos H. G., Stynes M., and Tobiska L., Numerical Methods for Singularly Perturbed Differential Equations, Convection-Diffusion and Flow problems, Springer, Berlin, 2008 | MR

[5] Zadorin A. I., Zadorin N. A., “Non-Polynomial Interpolation of Functions with Large Gradients and Its Application”, Comput. Math. Math. Phys., 61:2 (2021), 167–176 | DOI | DOI | MR

[6] Zadorin A. I., “Formulas for Numerical Differentiation of Functions with Large Gradients”, Numer. Anal. Appl., 16:1 (2023), 14–21 | DOI | DOI | MR

[7] Zadorin A.I., “Formulas for Numerical Differentiation on a Uniform Mesh in the Presence of a Boundary Layer”, Comput. Math. Math. Phys., 64:6 (2024), 1167–1175 | DOI | MR

[8] Zadorin A. I., “Application of a Taylor series to approximate a function with large gradients”, Sib. Electron. Math. Rep., 20:4 (2023), 1420–1429 | MR

[9] Ranjan R., Prasad H. S., “A novel approach for the numerical approximation to the solution of singularly perturbed differential-difference equations with small shifts”, J. of Applied Mathematics and Computing, 65 (2021), 403–427 | DOI | MR