×

Hermite spline interpolation on a three direction mesh from Powell-Sabin and Hsieh-Clough-Tocher finite elements. (English) Zbl 1357.41008

Summary: In this paper we develop a general local method to define Hermite interpolants of prescribed order \(r \geq 1\) and global class \(C^s\) on the three direction mesh of the real plane. They are defined from Powell-Sabin and Hsieh-Clough-Tocher finite elements in such a way that the interpolation operators have fundamental functions with compact support and reproduce a given space \(\mathbb{P}_m\) of polynomials included in the spline space.

MSC:

41A15 Spline approximation
41A10 Approximation by polynomials
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65D07 Numerical computation using splines
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Laghchim-Lahlou, M., Elements finis composites de classe \(C^k\) dans \(R^2 (1991)\), I.N.S.A. de Rennes, (Thèse)
[2] Laghchim-Lahlou, M.; Sablonnière, P., \(C^r\) finite element of HCT, PS and FVS types, (Gruber, R.; Périaux, J.; Shaw, R. P., Fifth International Symposium on Numerical Analysis in Engineering, Vol. 2 (1989), Springer-Verlag: Springer-Verlag Berlin), 163-168 · Zbl 0675.41015
[3] Laghchim-Lahlou, M.; Sablonnière, P., Composite \(C^r\) triangular finite elements of PS type on a three-direction mesh, (Laurent, P. J.; Le Méhauté, A.; Schumaker, L. L., Curves and Surfaces (1991), Academic Press: Academic Press New York), 275-278 · Zbl 0736.41003
[4] Laghchim-Lahlou, M.; Sablonnière, P., Eléments finis polynomiaux composés de classe \(C^r\), C. R. Acad. Sci. Paris Sér. I, t. 316, 503-508 (1993) · Zbl 0772.65006
[5] Laghchim-Lahlou, M.; Sablonnière, P., \(C^r\)-finite elements of Powell-Sabin type on the three direction mesh, Adv. Comput. Math., 6, 191-206 (1996) · Zbl 0867.65002
[6] Powell, M. J.D.; Sabin, M. A., Piecewise quadratic approximation on triangles, ACM Trans. Math. Softw., 3, 4, 316-325 (1977) · Zbl 0375.41010
[7] Sablonnière, P., Composite finite elements of class \(C^k\), J. Comput. Appl. Math., 12-13, 541-550 (1985) · Zbl 0587.41004
[8] Sablonnière, P., Elements finis triangulaires de degré \(5\) et de class \(C^2\), (Chenin, P.; Di Crescenzo, C.; Robert, F., Computers and Computing (1986), Wiley-Masson), 111-115
[9] Sablonnière, P., Composite finite elements of class \(C^2\), (Chui, C. K.; Schumaker, L. L.; Utreras, F. I., Topics in Multivariate Approximation (1987), Academic Press: Academic Press New York), 207-217
[10] Ciarlet, P. G., Interpolation error estimates for the reduced Hsieh-Clough-Tocher triangles, Math. Comp., 32, 335-344 (1978) · Zbl 0378.65010
[11] Ciarlet, P. G., Basic error estimates for elliptic problems, (Handbook of Numerical Analysis, Vol. II, Finite Element Methods, Part I (1991), North-Holland: North-Holland Amsterdam), 17-352 · Zbl 0875.65086
[13] Laghchim-Lahlou, M.; Sablonnière, P., Triangular Finite Elements of HCT type and class \(C^\rho \), Adv. Comput. Math., 2, 101-122 (1994) · Zbl 0832.65003
[14] Dahmen, W.; Goodman, T. N.T.; Micchelli, C. A., Local spline interpolation schemes in one and several variables, (Gómez, A.; Guerra, F.; Jiménez, M. A.; López, G., Approximation & Optimization, Proc. Havana 1987. Approximation & Optimization, Proc. Havana 1987, LNM, vol. 1354 (1987), Springer-Verlag), 11-24
[16] Schumaker, L. L.; Sorokina, T., Smooth macro-elements on Powell-Sabin-12 Splits, Mathemag. Comput., 75, 254, 711-726 (2005) · Zbl 1094.65012
[17] Davydov, O.; Sablonnière, P., \(C^2\) piecewise cubic quasi-interpolants on a 6-direction mesh, J. Approx. Theory, 162, 528-544 (2010) · Zbl 1193.41006
[18] Manni, C.; Sablonnière, P., Quadratic spline quasi-interpolants on Powell-Sabin partitions, Adv. Comput. Math., 26, 283-304 (2007) · Zbl 1116.65008
[19] Remogna, S., Bivariate \(C^2\) cubic spline quasi-interpolants on uniform Powell-Sabin triangulations of a rectangular domain, Adv. Comput. Math., 36, 39-65 (2012) · Zbl 1251.41001
[20] Sbibih, D.; Serghini, A.; Tijini, A.; Zidna, A., Superconvergent \(C^1\) cubic spline quasi-interpolants on Powell-Sabin partitions, BIT, 55, 797-821 (2015) · Zbl 1326.41005
[21] Speleers, H., A family of smooth quasi-interpolants defined over Powell-Sabin triangulations, Constr. Approx., 41, 297-324 (2015) · Zbl 1318.41007
[22] Slomson, A., An Introduction to Combinatorics (1991), Chapman and Hall · Zbl 0743.05002
[23] Van Lint, J. H.; Wilson, R. M., A Course in Combinatorics (2001), Cambridge University Press · Zbl 0980.05001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.