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SECTION A: EXACT SCIENCES

Vol. 6 No. 1 (2014)

On a class of embedded cubature formulae on the simplex

DOI
https://doi.org/10.18272/aci.v6i1.149
Submitted
September 29, 2015
Published
2014-06-13

Abstract

In this paper we investigate a class of embedded cubature formulae on the simplex announced in [1]. Here we recall the class of formulae, we introduce the remainder and we give an estimation of this, we also investigate the convergence. Some numerical examples are given.

Mathematical subject classification: Primary: 65D32

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References

  1. Costabile, F.; Guzzardi, L. 2009. "Extending univariate expansion formulae on the simplex". IJNMA, 1: 1-17.
  2. Stroud., A. 1971. "Approximate calculation of multiple integrals". Prentice-Hall, Inc.
  3. Lyness, J.; Cools, R. 1993. "A survey of numerical cubature over triangles". In A half-century of computational mathematics, Proc. Sympos. Appl. Math, 48: 1943-1993.
  4. Cools, R.; Rabinowitz, P. 1993. "Monomial Cubature Rules since Stroud: a compilation". J. Comput. Appl. Math, 48: 309-326.
  5. Cools, R. 2003. "An Encyclopedia of Cubature Formula". J. Complexity.
  6. Cools, R.; Haegemans, A. 1989. "On the construction of multi-dimensional embedded cubature formulae". Numer. Math, 55: 735-745.
  7. Costabile, F.; Dell"™Accio, F. 2001. "Expansions over a simplex of real functions by means of Bernoulli polynomials". Num. Alg, 28: 63-86.
  8. Costabile, F.; Dell"™Accio, F. 2007. "New embedded boundary-type quadrature formulas for the simplex". Numer Algor, 45: 253-267.
  9. Costabile, F.; Dell"™Accio, F.; Guzzardi, L. 2008. "New bi-variate polynomial expansions with boundary data on the simplex". Calcolo, 45: 177-192.
  10. Squire, W. 1961. "Some applications of quadrature by differentiation". J. Soc. Ind. Appl. Math, 9: 94-108.
  11. Varma, A. 1986. "On Birkhoff quadrature formulas". Proc. Amer. Math. Soc, 97: 38-40.
  12. Costabile, F.; Longo, E. 2010. "A Birkhoff interpolation problem and application". Calcolo, 47: 49-63.
  13. Kim, K.; Cools, R.; Ixaru, L. 2002. "Quadrature rules using first derivatives for oscillatory integrands". J. Comput. Appl. Math, 140: 479-497.
  14. Rajasekaran, S. 2007. "Symbolic computation and differential quadrature method - A boon to engineering analysis". Struct. Eng. Mech, 27: 713-739.
  15. Rathod, H.; Islam, M. 1998. "Integration of rational function of bi-variate polynomial numerators with linear denominator over a (-1,1) square in a local parametric two-dimensional space". Comput. Methods Appl. Mech. Engrg, 161: 195-213.
  16. Costabile, F. 1999. "Expansions of real functions in Bernoulli polynomials and applications". Conf. Sem., University Bari, 273.
  17. Costabile, F. F.and Dell"™Accio. 2001. "Expansion over a rectangle of real functions in Bernoulli polynomials and applications". BIT, 41: 451.
  18. http://mathworld.wolfram.com/bernoullipolynomial.html.
  19. Duffy, M. 1982. "Quadrature over a pyramid or cube of integrands with a singularity at a vertex". SIAM J. Numer. Anal., 19: 1260-1262.
  20. Engels, H. 1980. "Numerical Quadrature and Cubature". Academic Press.