Computational Mathematics and Mathematical Physics

, Volume 53, Issue 12, pp 1808–1818 | Cite as

Cubature formulas for a two-variable function with boundary-layer components

Article

Abstract

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.

Keywords

two-variable function boundary layer double integral nonpolynomial interpolation cubature rule error estimate 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    I. S. Berezin and N. P. Zhidkov, Computing Methods (Pergamon, Oxford, 1965; Nauka, Moscow, 1966).MATHGoogle Scholar
  2. 2.
    N. S. Bakhvalov, N. P. Zhidkov, and G. M. Kobel’kov, Numerical Methods (Nauka, Moscow, 1987) [in Russian].MATHGoogle Scholar
  3. 3.
    A. I. Zadorin and N. A. Zadorin, “Quadrature formulas for functions with a boundary-layer component,” Comput. Math. Math. Phys. 51, 1837–1846 (2011).CrossRefMathSciNetGoogle Scholar
  4. 4.
    G. I. Shishkin, Grid Approximations of Singularly Perturbed Elliptic and Parabolic Equations (Ural Otd. Ross. Akad. Nauk, Yekaterinburg, 1992) [in Russian].Google Scholar
  5. 5.
    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).CrossRefGoogle Scholar
  6. 6.
    H. G. Roos, M. Stynes, and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion and Flow Problems (Springer-Verlag, Berlin, 1996).CrossRefMATHGoogle Scholar
  7. 7.
    G. I. Shishkin and L. P. Shishkina, Difference Methods for Singular Perturbation Problems (Chapman and Hall/CRC, Boca Raton, 2009).MATHGoogle Scholar
  8. 8.
    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).MathSciNetGoogle Scholar
  9. 9.
    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).CrossRefMathSciNetGoogle Scholar
  10. 10.
    A. I. Zadorin, “Spline interpolation of functions with a boundary layer component,” Int. J. Numer. Anal. Model. Ser. 2(2–3), 262–279 (2011).MATHMathSciNetGoogle Scholar
  11. 11.
    N. N. Kalitkin, Numerical Methods (Nauka, Moscow, 1978) [in Russian].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Omsk Branch of the Sobolev Institute of Mathematics, Siberian BranchRussian Academy of SciencesOmskRussia

Personalised recommendations