Numerical integration based on bivariate quadratic spline quasi-interpolants on Powell-Sabin partitions
Tóm tắt
In this paper we generate and study new cubature formulas based on spline quasi-interpolants in the space of quadratic Powell-Sabin splines on nonuniform triangulations of a polygonal domain in ℝ2. By using a specific refinement of a generic triangulation, optimal convergence orders are obtained for some of these rules. Numerical tests are presented for illustrating the theoretical results.
Tài liệu tham khảo
Atkinson, K., Chien, D.: Piecewise polynomial collocation for boundary integral equations. Reports on Computational mathematics, 29, Dept of mathematics, University of Iowa, Iowa City (1992)
Chui, C.K.: Multivariate splines. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 54. SIAM, Philadelphia (1988)
Dagnino, C., Lamberti, P.: Numerical evaluation of Cauchy principal value integrals based on local spline approximation operators. J. Comput. Appl. Math. 76, 231–238 (1996)
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. 16(4), 68–79 (2000)
Dagnino, C., Lamberti, P.: On the approximation power of bivariate quadratic C1 splines. J. Comput. Appl. Math. 131, 321–332 (2001)
Dagnino, C., Lamberti, P.: Some performances of local bivariate quadratic C1 quasi-interpolating splines on nonuniform type-2 triangulations. J. Comput. Appl. Math. 173(1), 21–37 (2005)
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. Comput. Math. Appl. 33(1/2), 59–67 (1997)
Dagnino, C., Santi, E.: Spline product quadrature rules for Cauchy singular integrals. J. Comput. Appl. Math. 33, 133–140 (1990)
Dagnino, C., Santi, E.: On the convergence of spline product quadratures for Cauchy principal value integrals. J. Comput. Appl. Math. 36, 181–187 (1991)
Dagnino, C., Demichelis, V., Santi, E.: Numerical integration based on quasi-interpolating splines. Computing 50, 149–163 (1993)
Dagnino, C., Demichelis, V., Santi, E.: An algorithm for numerical integration based on quasi-interpolating splines. Numer. Algorithms 5, 443–452 (1993)
de Boor, C.: B-form basics. In: Farin, G.E. (ed.) Geometric Modeling: Algorithms and New Trends, pp. 131–148. SIAM, Philadelphia (1987)
de Boor, C., Höllig, K., Riemenschneider, S.: Box Splines. Springer, New York (1993)
Diercks, P.: On calculating normalized Powell-Sabin B-splines. Comput. Aided Geom. Des. 15, 61–78 (1997)
Diercks, P., Van Leemput, S., Vermeire, T.: Algorithms for surface fitting using Powel-Sabin splines. IMA J. Numer. Anal. 12, 271–299 (1992)
Evans, G.: Practical Numerical Integration. Wiley, New York (1993)
Haber, S.: Numerical evaluation of multiple integrals. SIAM Rev. 12(4), 481–526 (1970)
Lamberti, P.: Numerical integration based on bivariate quadratic spline quasi-interpolants on bounded domains. BIT Numer. Math. 49(3), 565–588 (2009)
Manni, C., Sablonnière, P.: Quadratic spline quasi-interpolants on Powell-Sabin partitions. Adv. Comput. Math. 26, 283–304 (2007)
Pittaluga, G., Sacripante, L.: A numerical algorithm for cubature by bivariate splines on nonuniform partitions. Numer. Algorithms 28, 273–284 (2001)
Powell, M.J.D., Sabin, M.A.: Piecewise quadratic approximations on triangles. ACM Trans. Math. Softw. 3, 316–325 (1977)
Rabinowitz, P.: Numerical integration based on approximating splines. J. Comput. Appl. Math. 33, 73–83 (1990)
Sablonnière, P.: Error bounds for Hermite interpolation by quadratic splines on an α-triangulation. IMA J. Numer. Anal. 7, 495–508 (1987)
Sablonnière, P.: Quasi-interpolants splines et applications. Lecture Notes, Prépublication IRMAR 06-41, Rennes, October 2006
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.: Error estimate and extrapolation of a quadrature formula derived from a quartic spline quasi-interpolant. BIT Numer. Math. 50(4), 843–862 (2010)
Sbibih, D., Serghini, A., Tijini, A.: Polar forms and quadratic spline quasi-interpolants on Powell-Sabin partitions. Appl. Numer. Math. 59, 938–958 (2009)
Willemans, K., Dierckx, P.: Surface fitting using convex Powell-Sabin splines. J. Comput. Appl. Math. 56, 263–282 (1994)
Windmolders, J.: Powell-Sabin splines for computer aided design. Ph.D. Thesis, University of Leuven (2003)
Windmolders, J., Dierckx, P.: Subdivision of uniform Powell-Sabin splines. Comput. Aided Geom. Des. 16, 301–315 (1999)