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27 Dec 2013
Cubature formulas for a twovariable function with boundarylayer components
 A. I. Zadorin
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Cubature formulas for evaluating the double integral of a twovariable function with boundarylayer components are constructed and studied. Because of the boundarylayer components, the cubature formulas based on NewtonCotes formulas become considerably less accurate. Analogues of the trapezoidal and Simpson rules that are exact for the boundarylayer components are constructed. Error estimates for the constructed formulas are derived that are uniform in the gradients of the integrand in the boundary layers.
Original Russian Text © A.I. Zadorin, 2013, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2013, Vol. 53, No. 12, pp. 1997–2007.
 Berezin, I S, Zhidkov, N P (1965) Computing Methods. Pergamon, Oxford
 Bakhvalov, N S, Zhidkov, N P, Kobel’kov, G M (1987) Numerical Methods. Nauka, Moscow
 Zadorin, A I, Zadorin, N A (2011) Quadrature formulas for functions with a boundarylayer component. Comput. Math. Math. Phys. 51: pp. 18371846 CrossRef
 Shishkin, G I (1992) Grid Approximations of Singularly Perturbed Elliptic and Parabolic Equations. Ural Otd. Ross. Akad. Nauk, Yekaterinburg
 Miller, J J H, O’Riordan, E, Shishkin, G I (2012) Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions. World Scientific, Singapore CrossRef
 Roos, H G, Stynes, M, Tobiska, L (1996) Numerical Methods for Singularly Perturbed Differential Equations: ConvectionDiffusion and Flow Problems. SpringerVerlag, Berlin CrossRef
 Shishkin, G I, Shishkina, L P (2009) Difference Methods for Singular Perturbation Problems. Chapman and Hall/CRC, Boca Raton
 Zadorin, A I, Zadorin, N A (2011) Interpolation of functions with boundary layer components and its application to the twogrid method. Sib. Elektron. Mat. Izv. 8: pp. 247267
 Zadorin, A I, Zadorin, N A (2010) Spline interpolation on a uniform grid for functions with a boundarylayer component. Comput. Math. Math. Phys. 50: pp. 211223 CrossRef
 Zadorin, A I (2011) Spline interpolation of functions with a boundary layer component. Int. J. Numer. Anal. Model. Ser. 2: pp. 262279
 Kalitkin, N N (1978) Numerical Methods. Nauka, Moscow
 Title
 Cubature formulas for a twovariable function with boundarylayer components
 Journal

Computational Mathematics and Mathematical Physics
Volume 53, Issue 12 , pp 18081818
 Cover Date
 20131201
 DOI
 10.1134/S0965542513120130
 Print ISSN
 09655425
 Online ISSN
 15556662
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 twovariable function
 boundary layer
 double integral
 nonpolynomial interpolation
 cubature rule
 error estimate
 Industry Sectors
 Authors

 A. I. Zadorin ^{(1)}
 Author Affiliations

 1. Omsk Branch of the Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, ul. Pevtsova 13, Omsk, 644043, Russia