Cubature formulas for a two-variable function with boundary-layer components
- A. I. Zadorin
- … show all 1 hide
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
Cubature formulas for evaluating the double integral of a two-variable function with boundary-layer components are constructed and studied. Because of the boundary-layer components, the cubature formulas based on Newton-Cotes formulas become considerably less accurate. Analogues of the trapezoidal and Simpson rules that are exact for the boundary-layer components are constructed. Error estimates for the constructed formulas are derived that are uniform in the gradients of the integrand in the boundary layers.
- I. S. Berezin and N. P. Zhidkov, Computing Methods (Pergamon, Oxford, 1965; Nauka, Moscow, 1966).
- N. S. Bakhvalov, N. P. Zhidkov, and G. M. Kobel’kov, Numerical Methods (Nauka, Moscow, 1987) [in Russian].
- A. I. Zadorin and N. A. Zadorin, “Quadrature formulas for functions with a boundary-layer component,” Comput. Math. Math. Phys. 51, 1837–1846 (2011). CrossRef
- G. I. Shishkin, Grid Approximations of Singularly Perturbed Elliptic and Parabolic Equations (Ural Otd. Ross. Akad. Nauk, Yekaterinburg, 1992) [in Russian].
- J. J. H. Miller, E. O’Riordan, and G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions (World Scientific, Singapore, 2012). CrossRef
- H. G. Roos, M. Stynes, and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion and Flow Problems (Springer-Verlag, Berlin, 1996). CrossRef
- G. I. Shishkin and L. P. Shishkina, Difference Methods for Singular Perturbation Problems (Chapman and Hall/CRC, Boca Raton, 2009).
- A. I. Zadorin and N. A. Zadorin, “Interpolation of functions with boundary layer components and its application to the two-grid method,” Sib. Elektron. Mat. Izv. 8, 247–267 (2011).
- A. I. Zadorin and N. A. Zadorin, “Spline interpolation on a uniform grid for functions with a boundary-layer component,” Comput. Math. Math. Phys. 50, 211–223 (2010). CrossRef
- A. I. Zadorin, “Spline interpolation of functions with a boundary layer component,” Int. J. Numer. Anal. Model. Ser. 2(2–3), 262–279 (2011).
- N. N. Kalitkin, Numerical Methods (Nauka, Moscow, 1978) [in Russian].
- Cubature formulas for a two-variable function with boundary-layer components
Computational Mathematics and Mathematical Physics
Volume 53, Issue 12 , pp 1808-1818
- Cover Date
- Print ISSN
- Online ISSN
- Springer US
- Additional Links
- two-variable function
- boundary layer
- double integral
- nonpolynomial interpolation
- cubature rule
- error estimate
- 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