Abstract
The BS Hermite spline quasi-interpolation scheme is presented. It is related to the continuous extension of the BS linear multistep methods, a class of Boundary Value Methods for the solution of Ordinary Differential Equations. In the ODE context, using the numerical solution and the associated numerical derivative produced by the BS methods, it is possible to compute, with a local approach, a suitable spline with knots at the mesh points collocating the differential equation at the knots and having the same convergence order as the numerical solution. Starting from this spline, here we derive a new quasi-interpolation scheme having the function and the derivative values at the knots as input data. When the knot distribution is uniform or the degree is low, explicit formulas can be given for the coefficients of the new quasi-interpolant in the B-spline basis. In the general case these coefficients are obtained as solution of suitable local linear systems of size 2d×2d, where d is the degree of the spline. The approximation order of the presented scheme is optimal and the numerical results prove that its performances can be very good, in particular when suitable knot distributions are used.
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Brugnano, L., Trigiante, D.: Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordon and Breach, New York (1998)
de Boor, C.: Splines as linear combinations of B-splines. In: Lorentz, G.G., (eds.) Approximation Theory II, pp. 1–47. Academic Press, San Diego (1976)
de Boor, C.: A Practical Guide to Splines, revised edn. Springer, Berlin (2001)
de Boor, C., Fix, M.G.: Spline approximation by quasi-interpolants. J. Approx. Theory 8, 19–54 (1973)
Gaffney, P.W., Powell, M.J.D.: Optimal interpolation, in numerical analysis. In: Proc. 6th Biennal Dundee Conf., Univ. Dundee, Dundee, 1975. Lecture Notes in Math., vol. 506, pp. 90–99. Springer, Berlin (1976)
Lee, B.G., Lyche, T., Mørken, K.: Some examples of quasi-interpolants constructed from local spline projectors. In: Lyche, T., Schumaker, L.L. (eds.) Mathematical Methods for Curves and Surfaces: Oslo 2000, pp. 243–252. Vanderbilt University Press, Nashville (2001)
Lyche, T., Schumaker, L.L.: Local spline approximation. J. Approx. Theory 15, 294–325 (1975)
Lyche, T., Morken, K.: Spline methods, Draft. Institute of Informatics, University of Oslo (2008)
Mazzia, F., Pavani, R.: A class of symmetric methods for Hamiltonian systems. In: Carini, A., Piva, R. (eds.) Atti del XVIII Congresso dell’Associazione Italiana di Meccanica Teorica e Applicata, Brescia, 11–14 Settembre 2007. Starrylink, Brescia (2007)
Mazzia, F., Sestini, A., Trigiante, D.: B-spline multistep methods and their continuous extensions. SIAM J. Numer. Anal. 44(5), 1954–1973 (2006)
Mazzia, F., Sestini, A., Trigiante, D.: BS linear multistep methods on non-uniform meshes. J. Numer. Anal. Ind. Appl. Math. 1, 131–144 (2006)
Mazzia, F., Sestini, A., Trigiante, D.: High order continuous approximation for the top order methods. In: Simos, E., Psihoyios, G., Tsitouras, C. (eds.) Numerical Analysis and Applied Mathematics, pp. 611–613. American Institute of Physics, Melville (2007)
Mazzia, F., Sestini, A., Trigiante, D.: The continous extension of the B-spline linear multistep methods for BVPs on non-uniform meshes. Appl. Numer. Meth. 59, 723–738 (2009)
Micchelli, C.A., Rivlin, T.J., Winograd, S.: The optimal recovery of smooth functions. Numer. Math. 26, 191–200 (1976)
Sablonnière, P.: Positive spline operators and orthogonal splines. J. Approx. Theory 52, 28–42 (1988)
Sablonnière, P.: Recent progress on univariate and multivariate polynomial and spline quasi-interpolants. In: de Bruin, M.G., Mache, D.H., Szabados, J. (eds.) Trends and Applications in Constructive Approximation. International Series of Numerical Mathematics, vol. 151, pp. 229–245. Birkhäuser, Basel (2005)
Sablonnière, P.: Univariate spline quasi-interpolants and applications to numerical analysis. Rend. Semin. Mat. Univ. (Torino) 63(3), 211–222 (2005)
Sablonnière, P., Sbibih, D.: Integral spline operators exact on polynomials. Approx. Theory Appl. 10(3), 56–73 (1994)
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Communicated by Tom Lyche.
Work developed within the project “Numerical methods and software for differential equations”.
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Mazzia, F., Sestini, A. The BS class of Hermite spline quasi-interpolants on nonuniform knot distributions. Bit Numer Math 49, 611–628 (2009). https://doi.org/10.1007/s10543-009-0229-9
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DOI: https://doi.org/10.1007/s10543-009-0229-9