Tensor product Gauss-Lobatto points are Fekete points for the cube

Bos, L.; Taylor, M.; Wingate, B.

doi:10.1090/S0025-5718-00-01262-X

Tensor product Gauss-Lobatto points are Fekete points for the cube
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by L. Bos, M. A. Taylor and B. A. Wingate PDF

Math. Comp. 70 (2001), 1543-1547 Request permission

Abstract:

Tensor products of Gauss-Lobatto quadrature points are frequently used as collocation points in spectral element methods. Unfortunately, it is not known if Gauss-Lobatto points exist in non-tensor-product domains like the simplex. In this work, we show that the $n$-dimensional tensor-product of Gauss-Lobatto quadrature points are also Fekete points. This suggests a way to generalize spectral methods based on Gauss-Lobatto points to non-tensor-product domains, since Fekete points are known to exist and have been computed in the triangle and tetrahedron. In one dimension this result was proved by Fejér in 1932, but the extension to higher dimensions in non-trivial.

References

L. Bos, On certain configurations of points in $\textbf {R}^n$ which are unisolvent for polynomial interpolation, J. Approx. Theory 64 (1991), no. 3, 271–280. MR 1094439, DOI 10.1016/0021-9045(91)90063-G
Chen, Q., and I. Babuška, Approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle, Comput. Methods Appl. Mech. Engrg., 128, 405–417, 1995.
Moshe Dubiner, Spectral methods on triangles and other domains, J. Sci. Comput. 6 (1991), no. 4, 345–390. MR 1154903, DOI 10.1007/BF01060030
Fejér, L., Bestimmung derjenigen Abszissen eines Intervalles für welche die Quadratsumme der Grundfunktionen der Lagrangeschen Interpolation im Intervalle $[-1,1]$ ein möglichst kleines Maximum besitzt, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mt. Ser. II, 1, 263–276, 1932.
Hesthaven, J. S., and C. H. Teng, Stable spectral methods on tetrahedral elements, SIAM J. Sci. Comput., 1999, in press.
Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. MR 832183, DOI 10.1017/CBO9780511810817
Komatitsch, D., et al., Wave propagation in 2-D elastic media using a spectral element method with triangles and quadrangles, submitted J. Comput. Acoust., 1999.
Maday, Y. and A. T. Patera, Spectral element methods for the incompressible Navier-Stokes equations, in State of the Art Surveys in Computational Mechanics, edited by A. K. Noor, ASME, New York, 1988.
Patera, A.T., A spectral element method for fluid dynamics: Laminar flow in a channel expansion, J. Comput. Phys., 54, 468–488, 1984.
S. J. Sherwin and G. E. Karniadakis, A triangular spectral element method; applications to the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 123 (1995), no. 1-4, 189–229. MR 1339373, DOI 10.1016/0045-7825(94)00745-9
Taylor, M. A., and B. A. Wingate, The Fekete collocation points for triangular spectral elements, SIAM J. Numer. Anal., 1998, submitted.
Taylor, M. A., and B. A. Wingate, A generalized diagonal mass matrix spectral element method for non-quadrilateral elements, Appl. Num. Math., 1999, in press.
Wingate, B. A., and J. P. Boyd, Spectral element methods on triangles for geophysical fluid dynamics problems, in Proceedings of the Third International Conference on Spectral and High-order Methods, edited by A. V. Ilin and L. R. Scott, pp. 305–314, Houston J. Mathematics, Houston, Texas, 1996.