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Multilevel quadratic spline quasi-interpolation. (English) Zbl 1433.65017
Summary: In this paper, we present new approximation schemes based on both quasi-interpolation and multilevel methods by using bivariate quadratic B-spline functions, defined on simple and multiple knot type-2 triangulations, improving the classical quasi-interpolating spline results. We also prove polynomial reproduction, optimal approximation order and propose some numerical results and applications.
65D07 Numerical computation using splines
41A15 Spline approximation
41A25 Rate of convergence, degree of approximation
65D15 Algorithms for approximation of functions
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
65D05 Numerical interpolation
Full Text: DOI
[1] de Boor, C.; Höllig, K.; Riemenschneider, S., Box Splines (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0814.41012
[2] Chui, C. K., Multivariate splines, Proceedings of the CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 54 (1988), SIAM: SIAM Philadelphia · Zbl 0687.41018
[3] Wang, R. H., Multivariate Spline Functions and Their Applications (2001), Kluwer: Kluwer Dordrecht
[4] Li, C.-Y.; Zhu, C. G., A multilevel univariate cubic spline quasi – interpolation and application to numerical integration, Math. Methods Appl. Sci., 33, 1578-1586 (2010) · Zbl 1193.41008
[5] Wang, R.-H.; Wu, J.; Zhang, X., Numerical integration based on multilevel quartic quasi-interpolants operator, Appl. Math. Comput., 227, 132-138 (2014) · Zbl 1364.65059
[6] Chui, C. K.; Wang, R. H., On a bivariate B-spline basis, Sci. Sin., XXVII, 1129-1142 (1984) · Zbl 0559.41010
[7] Dagnino, C.; Remogna, S.; Sablonnière, P., Error bounds on the approximation of functions and partial derivatives by quadratic spline quasi-interpolants on non-uniform criss-cross triangulations of a rectangular domain, BIT Numer. Math., 53, 87-109 (2013) · Zbl 1281.65019
[8] Foucher, F.; Sablonnière, P., Approximating partial derivatives of first and second order by quadratic spline quasi-interpolants on uniform meshes, Math. Comput. Simul., 77, 202-208 (2008) · Zbl 1135.65315
[9] Foucher, F.; Sablonnière, P., Superconvergence properties of some bivariate c^1 quadratic spline quasi-interpolants, (Cohen, A.; Merrien, J. L.; Schumaker, L. L., Curve and Surface Fitting: Avignon 2006 (2007), Nashboro Press: Nashboro Press Brentwood, TN), 160-169 · Zbl 1130.65013
[10] Wang, R.-H.; Lu, Y., Quasi-interpolating operators and their applications in hypersingular integrals, J. Comput. Math., 16, 4, 337-344 (1998) · Zbl 0920.65007
[11] Lamberti, P., Numerical integration based on bivariate quadratic spline quasi-interpolants on bounded domains, BIT Numer. Math., 49, 565-588 (2009) · Zbl 1181.65042
[12] Sablonnière, P., Quadratic spline quasi-interpolants on bounded domains of \(\mathbb{R}^d , d = 1 , 2 , 3\), Rend. Sem. Mat. Univ. Pol. Torino, 61, 3, 229-246 (2003) · Zbl 1121.41008
[13] Dagnino, C.; Lamberti, P., Numerical integration of 2-d integrals based on local bivariate c^1 quasi-interpolating splines, Adv. Comput. Math., 8, 1-2, 19-31 (1998) · Zbl 0893.65009
[14] Foucher, F.; Sablonnière, P., Approximating partial derivatives of first and second order by quadratic spline quasi-interpolants on uniform meshes, Math. Comput. Simul., 77, 202-208 (2008) · Zbl 1135.65315
[15] C. Dagnino, S. Remogna, Numerical Solution of Surface Integral Equations Based on Spline Quasi-interpolation, in: Proceedings of the Seventeenth International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2017, 4-8 July 2017, ISBN: 978-84-617-8694-7, pp. 695-703. · Zbl 1369.65173
[16] Allouch, C.; Sablonnière, P.; Sbibih, D., A collocation method for the numerical solution of a two dimensional integral equation using a quadratic spline quasi-interpolant, Numer. Alg., 62, 445-468 (2013) · Zbl 1266.65206
[17] Yu, R.-G.; Wang, R.-H.; Zhu, C. G., A numerical method for solving KDV equation with multilevel b-spline quasi-interpolation, Appl. Anal., 92, 8, 1682-1690 (2013) · Zbl 1308.65148
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