From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex

SIAM Journal on Numerical Analysis

Volumn 35, Issue 2, 1998, Pages 655-676

  Hesthaven, J S ^a  

^a BROWN UNIVERSITY   (United States)

Author keywords

Jacobi polynomials; Lebesgue constants; Polynomial interpolation; Spectral methods; Triangular elements

Indexed keywords

EID: 0000185847     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S003614299630587X     Document Type: Article

References (32)

1. T. J. STIELTJES, Sur quelques théorèmes d'algèbre. Comptes Rendus de l'Acadèmie des Sciences, 100 (1885), pp. 439-440.
2. T. J. STIELTJES, Sur les Polynômes de Jacobi, Comptes Rendus Acad. Sci., 100 (1885), pp. 620-622.
3. G. SZEGÖ. Orthogonal Polynomials, American Mathematical Society, 5th ed., Providence, RI, 1975.
4. C. CANUTO, M. Y. HUSSAINI, A. QUARTERONI, AND T. A. ZANG, Spectral Methods in Fluid Mechanics. Springer Series in Computational Physics, Springer-Verlag, Berlin, 1988.
5. I. BABUŠKA AND M. SURI, The p- and h-p version of the finite element method. An overview, Comput. Methods Appl. Mech. Engrg., 80 (1990), pp. 5-26.
6. A. PATERA, A spectral element method for fluid dynamics: Laminar flow in a channel, J. Comput. Phys., 54 (1984), pp. 468-488.
7. R. P. FEINERMAN AND D. J. NEWMAN, Polynomial Approximation, Williams and Wilkins, Baltimore, MD, 1974.
8. K. CHUNG AND T. YAO, On lattices admitting unique Lagrange interpolations, SIAM J. Numer. Anal., 14 (1977), pp. 735-743.
9. N. J. HIGHAM, Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia, PA, 1996.
10. P. J. DAVIS, Interpolation and Approximation, Dover Publications, New York, 1975.
11. T. J. RIVLIN, An Introduction to the Approximation of Functions, Dover, New York, 1969.
12. J. SZABADOS AND P. VÉRTESI, Interpolation of Functions, World Scientific, River Edge, NJ, 1990.
13. F. W. LUTTMANN AND T. J. RIVLIN, Some numerical experiments in the theory of polynomial interpolation, IBM J. Res. Develop., 9 (1965), pp. 187-191.
14. T. A. KILGORE, A characterization of the Lagrange interpolation projection with minimal Tchebycheff norm, J. Approx. Theory, 24 (1978), pp. 273-288.
15. C. DE BOOR AND A. PINKUS, Proof of the conjecture of Bernstein and Erdos concerning the optimal nodes for polynomial interpolation, J. Approx. Theory, 24 (1978), pp. 289-303.
16. J. R. ANGELOS, E. H. KAUFMAN JR., M. S. HENRY, AND T. D. LENKER, Optimal Nodes for Polynomial Interpolation, in Approximation Theory VI: Vol I. C. K. Chui, L. L. Schumaker, and J. D. Ward, eds., Academic Press, New York, 1989. pp. 17-20.
17. P. ERDÖS, Problem and Results on the Theory of Interpolation. II. Acta Math. Acad. Sci. Hungar., 12 (1961), pp. 235-244.
18. A. H. TURETSKII, The bounding of polynomials prescribed at equally distributed points, Proc. Pedag. Inst. Vitebsk., 3 (1940), pp. 117-127.
19. A. SCHÖNHAGE, Fehlerfortpflanzung bei Interpolation, Numer. Math., 3 (1961), pp. 62-71.
20. L. N. TREFETHEN AND J. C. WEIDEMAN, Two results on polynomial interpolation in equally spaced points, J. Approx. Theory, 65 (1991), pp. 247-260.
21. R. GÜNTTNER, Evaluation of Lebesgue constants, SIAM J. Numer. Anal., 17 (1980), pp. 512-520.
22. P. N. SHIVAKUMAR AND R. WONG, Asymptotic expansion of the Lebesgue constants associated with polynomial interpolation, Math. Comp., 39 (1982), pp. 195-200.
23. A. ERDÉLYI, Higher transcendental functions, Vol. II, Robert Krieger, Melbourne, FL, 1953.
24. J. H. MCCABE AND G. M. PHILLIPS, On a Certain Class of Lebesgue Constants, BIT, 13 (1973), pp. 434-442.
25. L. FEJÉR, Lagrangesche Interpolation und die Zugehörigen Konjugierten Punkte, Math. Ann., 106 (1932), pp. 1-55.
26. B. SUNDERMANN, Lebesgue Constants in Lagrangian Interpolation at the Fekete Points, Ergebnisbericht der Lehrstuhle Math III, VIII, 40, Univ. Dortmurm, Germany, 1980.
27. Q. CHEN AND I. BABUŠKA, Approximate optimal points for polynomial interpolation of real functions in an internal and in a triangle, Comput. Methods Appl. Mech. Engrg., 128 (1995), pp. 405-417.
28. L. BRUTMAN, On the Lebesgue function for polynomial interpolation, SIAM J. Numer. Anal., 15 (1978), pp. 694-704.
29. L. P. Bos, Bounding the Lebesgue function for interpolation in a simplex, J. Approx. Theory, 38 (1983), pp. 43-59.
30. E. HAIRER, S. P. NØRSETT, AND G. WANNER, Solving Ordinary Differential Equations I. Springer Series in Computational Mathematics 8, Springer-Verlag, Berlin, 1980.
31. J. J. MORE, B. S. GARBOW, AND K. E. HILLSTROM, User Guide for MINPACK-l, Report ANL-80-74, Argonne National Laboratory, Argonne, IL, 1980.
32. J. S. HESTHAVEN AND D. GOTTLIEB, Stable spectral methods for conservations laws on triangles with unstructured grids, SIAM J. Numer. Anal., 1997, submitted.