Abstract
An open question concerns the spatial distribution of nodes that are suitable for high-order Lagrange interpolation on the triangle and tetrahedron. Several current methods used to produce nodal sets with small Lebesgue constants are recalled. A new approach is presented for building nodal distributions of arbitrary order, that is based on curvilinear finite-element techniques. Numerical results are shown which demonstrate that, despite the explicit nature of this construction, the resulting node sets are well suited for interpolation and competitive with existing sets for up to tenth-order polynomial interpolation. Matlab scripts which evaluate the node distributions on the equilateral triangle are included.
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References
Bos LP (1983) Bounding the Lebesgue function for Lagrange interpolation in a simplex. J Approx Theory 38:43–59
Fekete M (1923) Über die verteilung der wurzeln bei gewissen algebraischen gleichungen mit ganzzahligen koeffizienten. Math Zeit 17:228–249
Chen Qi, Babuska Ivo (1995) Approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle. Comp Meth App Mech Eng 128:405–417
Taylor M, Wingate B, Vincent RE (2000) An algorithm for computing fekete points in the triangle. SIAM J Num Anal 38:1707–1720
Heinrichs W (2005) Improved Lebesgue constants on the triangle. J Comp Phys 207:625–638
Roth MJ (2005) Nodal configurations and Voronoi Tessellations for triangular spectral elements. PhD thesis, University of Victoria
Hesthaven JS (1998) From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex. SIAM J Numer Anal 35:655–676
Stieltjes TJ (1885) Sur quelques théorèmes d’algèbre. Comptes Rendus de l’Academie des Sciences 100:439–440
Stieltjes TJ (1885) Sur les polynômes de jacobi. Comptes Rendus Acad Sci 100:620–622
Blyth MG, Pozrikidis C (2005) A Lobatto interpolation grid over the triangle. IMA J Appl Math 153–169
Bittencourt ML (2005) Fully tensorial nodal and modal shape functions for triangles and tetrahedra. Int J Numer Meth Eng 63:1530–1558
Gordon WN, Hall CA (1973) Construction of curvilinear coordinate systems and application to mesh generation. Int J Num Meth Eng 7:461–477
Proriol J (1957) Sur une Famille de Polynomes à deux Variables Orthogonaux dans un Triangle. C R Acad Sci Paris 257:2459–2461
Forsythe GE, Malcolm MA, Moler CB (1976) Computer methods for mathematical computations. Prentice-Hall
Brent RP (1973) Algorithms for minimization without derivatives. Prentice-Hall
Hesthaven JS, Teng CH (2000) Stable spectral methods on tetrahedral elements. SIAM J Sci Comp 21:2352–2380
Chen Qi, Babuska Ivo (1996) The optimal symmetrical points for polynomial interpolation of real functions in the tetrahedron. Comp Meth App Mech Eng 137:89–94
Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in C: The art of computing scientific, 2nd edn. Cambridge University Press
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Warburton, T. An explicit construction of interpolation nodes on the simplex. J Eng Math 56, 247–262 (2006). https://doi.org/10.1007/s10665-006-9086-6
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DOI: https://doi.org/10.1007/s10665-006-9086-6