Quadrature formulas for functions with a boundary-layer component

Задорин, А. И.; Задорин, Н. А.

doi:10.1134/s0965542511110157

Cited by 11 publications

(9 citation statements)

References 4 publications

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“…It is a two dimensional version of the interpolation discussed in §2 of [7]. In Table 3 we present the numerical results using scheme (2.2) for test problem (3.1) and we observe that the numerical method is globally convergent for this example when the specially designed interpolation (5.1) is used to generate a global approximation to the solution.…”

Section: Interpolation Based On the Error Functionmentioning

confidence: 94%

A Singularly Perturbed Reaction-Diffusion Problem with Incompatible Boundary-Initial Data

Gracia

O’Riordan

2013

Lecture Notes in Computer Science

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Abstract.A singularly perturbed reaction-diffusion parabolic problem with an incompatibility between the initial and boundary conditions is examined. A finite difference scheme is considered which utilizes a special finite difference operator and a piecewise uniform Shishkin mesh. Numerical results are presented for both nodal and global pointwise convergence, using bilinear interpolation and, also, an interpolation method based on the error function. These results show that the method is not globally convergent when bilinear interpolation is used but they indicate that, for the test problem considered, it is globally convergent using the second type of interpolation.

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Section: Interpolation Based On the Error Functionmentioning

confidence: 94%

A Singularly Perturbed Reaction-Diffusion Problem with Incompatible Boundary-Initial Data

Gracia

O’Riordan

2013

Lecture Notes in Computer Science

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“…It is well known that in the regular case of an integrand with bounded derivatives, there is a composite formula based on the Newton-Cotes formula with m nodes that has an error not greater than O(h m ). In papers [10][11][12] it is shown that if the function u(x) has a boundary layer component corresponding to the representation (1.1), the error of the composite quadrature formulas based on Newton-Cotes formulas with a number of nodes m = 2, 3, 4, 5 becomes a quantity of order O(h) for ε ≤ h. Thus, the error of the Newton-Cotes formulas increases considerably as the parameter ε decreases.…”

Section: Newton-cotes Formulas On a Shishkin Gridmentioning

confidence: 99%

“…To increase the accuracy, in papers [10][11][12] analogs of Newton-Cotes formulas are constructed on a uniform grid with 2-5 nodes for the quadrature formulas to be exact for a boundary layer component known up to a factor. The quadrature formulas are constructed by approximating the integrand by an interpolant that is exact for the boundary layer component [6].…”

Section: Newton-cotes Formulas On a Shishkin Gridmentioning

confidence: 99%

“…The quadrature formulas are constructed by approximating the integrand by an interpolant that is exact for the boundary layer component [6]. It is proved in [10][11][12] that if the thus constructed quadrature formula has m nodes and the (m − 1)th derivative of the regular component is bounded, the composite formula has an error of order O(h m−1 ) uniformly in the boundary layer component and its derivatives.…”

Section: Newton-cotes Formulas On a Shishkin Gridmentioning

confidence: 99%

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

Задорин

2015

Numer. Analys. Appl.

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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.

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“…In [7] is offered to build quadrature formulas for functions of the form (1) which are exact on the boundary layer component Φ(x). For this purpose the function under the integral was replaced by interpolant (3), as a result the quadrature formula was constructed.…”

Section: Construction Of the Quadrature Formulamentioning

confidence: 99%

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

Задорин¹

2016

Model. anal. inf. sist.

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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.

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Quadrature formulas for functions with a boundary-layer component

Cited by 11 publications

References 4 publications

A Singularly Perturbed Reaction-Diffusion Problem with Incompatible Boundary-Initial Data

A Singularly Perturbed Reaction-Diffusion Problem with Incompatible Boundary-Initial Data

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

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