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Calcolo

, 55:4 | Cite as

Cubature formulae for nearly singular and highly oscillating integrals

  • Donatella Occorsio
  • Giada Serafini
Article
  • 71 Downloads

Abstract

The paper deals with the approximation of integrals of the type
$$\begin{aligned} I(f;{\mathbf {t}})=\int _{{\mathrm {D}}} f({\mathbf {x}}) {\mathbf {K}}({\mathbf {x}},{\mathbf {t}}) {\mathbf {w}}({\mathbf {x}}) d{\mathbf {x}},\quad \quad {\mathbf {x}}=(x_1,x_2),\quad {\mathbf {t}}\in \mathrm {T}\subseteq \mathbb {R}^p, \ p\in \{1,2\} \end{aligned}$$
where \({\mathrm {D}}=[-\,1,1]^2\), f is a function defined on \({\mathrm {D}}\) with possible algebraic singularities on \(\partial {\mathrm {D}}\), \({\mathbf {w}}\) is the product of two Jacobi weight functions, and the kernel \({\mathbf {K}}\) can be of different kinds. We propose two cubature rules determining conditions under which the rules are stable and convergent. Along the paper we diffusely treat the numerical approximation for kernels which can be nearly singular and/or highly oscillating, by using a bivariate dilation technique. Some numerical examples which confirm the theoretical estimates are also proposed.

Keywords

Cubature rules Orthogonal polynomials Approximation by polynomials 

Mathematics Subject Classification

65D32 41A05 41A10 

Notes

Acknowledgements

We want to thank the anonymous referee for the careful reading of the manuscript and for the valuable comments. We are also grateful to Professor G. Mastroianni for his helpful suggestions.

References

  1. 1.
    Caliari, M., De Marchi, S., Sommariva, A., Vianello, M.: Padua2DM: fast interpolation and cubature at the Padua points in Matlab/Octave. Numer. Algorithms 56(1), 45–60 (2011)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2nd edn. Computer Science and Applied Mathematics. Academic Press Inc, Orlando, FL (1984)MATHGoogle Scholar
  3. 3.
    Da Fies, G., Sommariva, A., Vianello, M.: Algebraic cubature by linear blending of elliptical arcs. Appl. Numer. Math. 74, 49–61 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    De Bonis, M.C., Pastore, P.: A quadrature formula for integrals of highly oscillatory functions. Rend. Circ. Mat. Palermo (2) Suppl. 82, 279–303 (2010)MathSciNetGoogle Scholar
  5. 5.
    Dobbelaere, D., Rogier, H., De Zutter, D.: Accurate 2.5-D boundary element method for conductive media. Radio Sci. 49, 389–399 (2014)CrossRefGoogle Scholar
  6. 6.
    Gautschi, W.: On the construction of Gaussian quadrature rules from modified moments. Math. Comput. 24, 245–260 (1970)MathSciNetMATHGoogle Scholar
  7. 7.
    Huybrechs, D., Vandewalle, S.: The construction of cubature rules for multivariate highly oscillatory integrals. Math. Comput. 76(260), 1955–1980 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Johnston, B.M., Johnston, P.R., Elliott, D.: A sinh transformation for evaluating two dimensional nearly singular boundary element integrals. Int. J. Numer. Methods Eng. 69, 1460–1479 (2007)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Lewanowicz, S.: A fast algorithm for the construction of recurrence relations for modified moments, (English summary). Appl. Math. (Warsaw) 22(3), 359–372 (1994)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Mastroianni, G., Milovanović, G.V.: Interpolation Processes. Basic Theory and Applications. Springer Monographs in Mathematics. Springer, Berlin (2008)CrossRefMATHGoogle Scholar
  11. 11.
    Mastroianni, G., Milovanović, G.V., Occorsio, D.: A Nyström method for two variables Fredholm integral equations on triangles. Appl. Math. Comput. 219, 7653–7662 (2013)MathSciNetMATHGoogle Scholar
  12. 12.
    Mastronardi, N., Occorsio, D.: Product integration rules on the semiaxis. In: Proceedings of the Third International Conference on Functional Analysis and Approximation Theory, vol. II (Acquafredda di Maratea, 1996). Rend. Circ. Mat. Palermo (2) Suppl. No. 52, Vol. II 605–618 (1998)Google Scholar
  13. 13.
    Monegato, G., Scuderi, L.: A polynomial collocation method for the numerical solution of weakly singular and singular integral equations on non-smooth boundaries. Int. J. Methods Eng. 58, 1985–2011 (2003)CrossRefMATHGoogle Scholar
  14. 14.
    Nevai, P.: Mean convergence of Lagrange Interpolation. III. Trans. Am. Math. Soc. 282(2), 669–698 (1984)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Occorsio, D., Russo, M.G.: Numerical methods for Fredholm integral equations on the square. Appl. Math. Comput. 218, 2318–2333 (2011)MathSciNetMATHGoogle Scholar
  16. 16.
    Pastore, P.: The numerical treatment of Love’s integral equation having very small parameter. J. Comput. Appl. Math. 236, 1267–1281 (2011)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Piessens, R.: Modified Clenshaw–Curtis integration and applications to numerical computation of integral transforms, Numerical integration (Halifax, N.S., 1986), NATO Advanced Science Institutes Series C, Mathematical and Physical Sciences, vol. 203, pp. 35–51. Reidel, Dordrecht (1987)Google Scholar
  18. 18.
    Serafini, G.: Numerical approximation of weakly singular integrals on a triangle. In: AIP Conference Proceedings 1776, 070011.  https://doi.org/10.1063/1.4965357 (2016)
  19. 19.
    Sloan, I.H.: Polynomial interpolation and hyperinterpolation over general regions. J. Approx. Theory 83(2), 238–254 (1995)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Van Deun, J., Bultheel, A.: Modified moments and orthogonal rational functions. Appl. Numer. Anal. Comput. Math. 1(3), 455–468 (2004)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Xu, Y.: On Gauss–Lobatto integration on the triangle. SIAM J. Numer. Anal. 49(2), 541–548 (2011)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Computer Sciences and EconomicsUniversity of BasilicataPotenzaItaly

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