Spline interpolation on a uniform grid for functions with a boundary-layer component

Задорин, А. И.; Задорин, Н. А.

doi:10.1134/s0965542510020028

Cited by 17 publications

(8 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The solution to a singular perturbation boundary value problem may have boundary-layer regions with large gradients [2,3]. According to [4,5], the use of a Lagrange polynomial to interpolate a function with a rapidly changing component corresponding to an exponential boundary layer may lead to errors of O (1). To increase the accuracy of interpolation of functions with boundary layer components, an analog of the Lagrange interpolation polynomial that is exact for these components is constructed in [6].…”

Section: Introductionmentioning

confidence: 99%

Lagrange interpolation and Newton-Cotes formulas for functions with boundary layer components on piecewise-uniform grids

Задорин

2015

Numer. Analys. Appl.

Self Cite

View full text Add to dashboard Cite

The problem of interpolation of a one-variable function considered as a solution to a boundary value problem for an equation with a small parameter ε in the highest derivative is investigated. The application of a Lagrange polynomial for such a function on a uniform grid may result in serious errors. For a Shishkin grid, ε-uniform error estimates of Lagrange polynomial interpolation are obtained. The Shishkin grid is modified to increase the interpolation accuracy. For Newton-Cotes formulas on such grids, ε-uniform error estimates are obtained. The results of numerical experiments confirm the theoretical estimates.

show abstract

Section: Introductionmentioning

confidence: 99%

Lagrange interpolation and Newton-Cotes formulas for functions with boundary layer components on piecewise-uniform grids

Задорин

2015

Numer. Analys. Appl.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Conditions (6) are satisfied in a case of function Φ(x) = e (x−1)/ε , corresponding to exponential boundary layer [3].…”

Section: Interpolation Formula and Its Propertiesmentioning

confidence: 99%

Interpolation Formulas for Functions with Large Gradients in the Boundary Layer and their Application

Задорин¹

2016

Model. anal. inf. sist.

Self Cite

View full text Add to dashboard Cite

Abstract. Interpolation of functions on the basis of Lagrange's polynomials is widely used. However in the case when the function has areas of large gradients, application of polynomials of Lagrange leads to essential errors. It is supposed that the function of one variable has the representation as a sum of regular and boundary layer components. It is supposed that derivatives of a regular component are bounded to a certain order, and the boundary layer component is a function, known within a multiplier; its derivatives are not uniformly bounded. A solution of a singularly perturbed boundary value problem has such a representation. Interpolation formulas, which are exact on a boundary layer component, are constructed. Interpolation error estimates, uniform in a boundary layer component and its derivatives are obtained. Application of the constructed interpolation formulas to creation of formulas of the numerical differentiation and integration of such functions is investigated.Keywords: function of one variable, boundary layer component, nonpolynomial interpolation, quadrature formulas, formulas of numerical differentiation, error estimate.

show abstract

“…The aim of this paper is investigation of the numerical differentiation formulas on a uniform mesh for functions with large gradients based on interpolation formulas from [29][30][31][32]. We consider the case of exponential boundary layer for the first and the second derivatives of such function with two or three interpolation nodes.…”

Section: Introductionmentioning

confidence: 99%

“…However, it is known that these polynomials on a uniform mesh as applied to the interpolation of functions with boundary layer components and its derivatives lead to errors of order O (1), see [26][27][28][29][30][31][32][33][34]37] and the references therein. To obtain the estimates that are uniform with respect to small parameter, we can use polynomial interpolation on a fitted mesh like the piecewise-uniform Shishkin mesh or construct on a uniform mesh the interpolation formula that is exact on boundary layer components.…”

Section: Introductionmentioning

confidence: 99%

“…To obtain the estimates that are uniform with respect to small parameter, we can use polynomial interpolation on a fitted mesh like the piecewise-uniform Shishkin mesh or construct on a uniform mesh the interpolation formula that is exact on boundary layer components. In this paper we investigate the numerical differentiation formulas for the functions with large gradients based on the interpolation formulas that are exact on boundary layer components on a uniform mesh, which were proposed in [29][30][31][32].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Analysis of the numerical differentiation formulas of functions with large gradients

Tikhovskaya

2017

AIP Conference Proceedings

View full text Add to dashboard Cite

A two-grid method with Richardson extrapolation for a semilinear convection-diffusion problem AIP Conference Proceedings 1684, 090007 (2015) Abstract. The solution of a singularly perturbed problem corresponds to a function with large gradients. Therefore the question of interpolation and numerical differentiation of such functions is relevant. The interpolation based on Lagrange polynomials on uniform mesh is widely applied. However, it is known that the use of such interpolation for the function with large gradients leads to estimates that are not uniform with respect to the perturbation parameter and therefore leads to errors of order O(1). To obtain the estimates that are uniform with respect to the perturbation parameter, we can use the polynomial interpolation on a fitted mesh like the piecewise-uniform Shishkin mesh or we can construct on uniform mesh the interpolation formula that is exact on the boundary layer components. In this paper the numerical differentiation formulas for functions with large gradients based on the interpolation formulas on the uniform mesh, which were proposed by A.I. Zadorin, are investigated. The formulas for the first and the second derivatives of the function with two or three interpolation nodes are considered. Error estimates that are uniform with respect to the perturbation parameter are obtained in the particular cases. The numerical results validating the theoretical estimates are discussed.

show abstract

Spline interpolation on a uniform grid for functions with a boundary-layer component

Cited by 17 publications

References 4 publications

Lagrange interpolation and Newton-Cotes formulas for functions with boundary layer components on piecewise-uniform grids

Lagrange interpolation and Newton-Cotes formulas for functions with boundary layer components on piecewise-uniform grids

Interpolation Formulas for Functions with Large Gradients in the Boundary Layer and their Application

Analysis of the numerical differentiation formulas of functions with large gradients

Contact Info

Product

Resources

About