Abstract
In this paper we consider the space generated by the scaled translates of the trivariate C 2 quartic box spline B defined by a set X of seven directions, that forms a regular partition of the space into tetrahedra. Then, we construct new cubature rules for 3D integrals, based on spline quasi-interpolants expressed as linear combinations of scaled translates of B and local linear functionals.
We give weights and nodes of the above rules and we analyse their properties.
Finally, some numerical tests and comparisons with other known integration formulas are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alayliouglu, A., Lubinsky, D.S., Eyre, D.: Product integration of logarithmic singular integrands based on cubic splines. J. Comput. Appl. Math. 11, 353–366 (1984)
Bojanov, B.D., Hakopian, H.A., Sahakian, A.A.: Spline Functions and Multivariate Interpolation. Kluwer, Dordrecht (1993)
Dagnino, C., Lamberti, P.: Numerical integration of 2-D integrals based on local bivariate C 1 quasi-interpolating splines. Adv. Comput. Math. 8, 19–31 (1998)
Dagnino, C., Lamberti, P.: Finite part integrals of local bivariate C 1 quasi-interpolating splines. Approx. Theory Appl. (N.S.) 16(4), 68–79 (2000)
Dagnino, C., Palamara Orsi, A.: Product integration of piecewise continuous integrands based on cubic spline interpolation at equally spaced nodes. Numer. Math. 52, 459–466 (1988)
Dagnino, C., Rabinowitz, P.: Product integration of singular integrands using quasi-interpolatory splines. Approximation theory and applications. Comput. Math. Appl. 33, 59–67 (1997)
Dagnino, C., Demichelis, V., Santi, E.: An algorithm for numerical integration based on quasi-interpolating splines. Numer. Algorithms 5, 443–452 (1993)
Dagnino, C., Demichelis, V., Santi, E.: Numerical integration based on quasi-interpolating splines. Computing 50, 149–163 (1993)
Dagnino, C., Perotto, S., Santi, E.: Convergence of rules based on nodal splines for the numerical evaluation of certain 2D Cauchy principal value integrals. J. Comput. Appl. Math. 89(2), 225–235 (1998)
Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2nd edn. Academic Press, New York (1984)
de Boor, C.: B-form basics. In: Farin, G.E. (ed.) Geometric Modeling: Algorithms and New Trends, pp. 131–148. Society for Industrial and Applied Mathematics, Philadelphia (1987)
de Boor, C., Höllig, K., Riemenschneider, S.: Box Splines. Springer, New York (1993)
Demichelis, V., Sablonnière, S.: Numerical integration by spline quasi-interpolants in three variables. In: Cohen, A., Merrien, J.L., Schumaker, L.L. (eds.) Curve and Surface Fitting, Avignon, 2006, pp. 131–140. Nashboro, Nashville (2007)
Demichelis, V., Sablonnière, S.: Cubature rule associated with a discrete blending sum of quadratic spline quasi-interpolants. J. Comput. Appl. Math. 235, 174–185 (2010)
Diethelm, K.: Uniform convergence of optimal order quadrature rules for Cauchy principal value integrals. J. Comput. Appl. Math. 56, 321–329 (1994)
Evans, G.A.: Practical Numerical Integration. Wiley, New York (1993)
Genz, A.C.: Testing multidimensional integration routines. In: Ford, B., Rault, J.C., Thomasset, F. (eds.) Tools, Methods and Languages for Scientific and Engineering Computation, pp. 81–94. North Holland, Amsterdam (1984)
Genz, A.C.: A package for testing multiple integration subroutines. In: Keast, P., Fairweather, G. (eds.) Numerical Integration: Recent Developments, Software and Applications, pp. 337–340. Reidel, Dordrecht (1987)
Greville, T.N.E.: Spline functions, interpolation and numerical quadrature. In: Ralston, A., Wilf, H.S. (eds.) Mathematical Methods for Digital Computers, Vol. II, pp. 156–168. Wiley, New York (1967)
Kim, M., Peters, J.: Fast and stable evaluation of box-splines via the BB-form. Numer. Algorithms 50(4), 381–399 (2009)
Krylov, V.I.: Approximate Calculation of Integrals. Macmillan, New York (1962)
Lamberti, P.: Numerical integration based on bivariate quadratic spline quasi-interpolants on bounded domains. BIT Numer. Math. 49, 565–588 (2009)
Li, C.J., Dagnino, C.: An adaptive numerical integration algorithm for polygons. Appl. Numer. Math. 60, 165–175 (2010)
Li, C.J., Lamberti, P., Dagnino, C.: Numerical integration over polygons using an eight-node quadrilateral spline finite element. J. Comput. Appl. Math. 233, 279–292 (2009)
Li, C.J., Demichelis, V., Dagnino, C.: Finite-part integrals over polygons by an 8-node quadrilateral spline finite element. BIT Numer. Math. 50, 377–394 (2010)
Peters, J.: C 2 surfaces built from zero sets of the 7-direction box spline. In: Mullineux, G. (ed.) IMA Conference on the Mathematics of Surfaces, pp. 463–474. Clarendon, Oxford (1994)
Rabinowitz, P.: Numerical integration based on approximating splines. J. Comput. Appl. Math. 33, 73–83 (1990)
Remogna, S.: Quasi-interpolation operators based on the trivariate seven-direction C 2 quartic box spline. BIT Numer. Math. 51(3), 757–776 (2011)
Sablonnière, P.: A quadrature formula associated with a univariate quadratic spline quasi-interpolant. BIT Numer. Math. 47, 825–837 (2007)
Sablonnière, P., Sbibih, D., Tahrichi, M.: Numerical integration based on bivariate quadratic spline quasi-interpolants on Powell-Sabin partitions. BIT Numer. Math. 53(1), 175–192 (2013)
Stroud, A.H.: Approximate Calculation of Multiple Integrals. Prentice-Hall Series in Automatic Computation. Prentice-Hall, Englewood Cliffs (1971)
Wang, R.H., Zhang, X.L.: Numerical integration based on bivariate quartic quasi-interpolation operators. Numer. Math. J. Chin. Univ. 16, 226–232 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Tom Lyche.
Rights and permissions
About this article
Cite this article
Dagnino, C., Lamberti, P. & Remogna, S. Numerical integration based on trivariate C 2 quartic spline quasi-interpolants. Bit Numer Math 53, 873–896 (2013). https://doi.org/10.1007/s10543-013-0431-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10543-013-0431-7