Optimal error estimates in Jacobi-weighted Sobolev spaces for polynomial approximations on the triangle

Li, Huiyuan; Shen, Jie

doi:10.1090/S0025-5718-09-02308-4

Optimal error estimates in Jacobi-weighted Sobolev spaces for polynomial approximations on the triangle
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by Huiyuan Li and Jie Shen PDF

Math. Comp. 79 (2010), 1621-1646 Request permission

Abstract:

Spectral approximations on the triangle by orthogonal polynomials are studied in this paper. Optimal error estimates in weighted semi-norms for both the $L^2-$ and $H^1_0-$orthogonal polynomial projections are established by using the generalized Koornwinder polynomials and the properties of the Sturm-Liouville operator on the triangle. These results are then applied to derive error estimates for the spectral-Galerkin method for second- and fourth-order equations on the triangle. The generalized Koornwinder polynomials and approximation results developed in this paper will be useful for many other applications involving spectral and spectral-element approximations in triangular domains.

References

Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
P. Appell. Sur des polynômes de deux variables analogues aux polynômes de Jacobi. Arch. Math. Phys., 66:238–245, 1881.
P. Appell and J. Kampeé de Fériet. Fonctions Hypergéométriques et Hypersphériques: Polyno͡mes d’Hermite. Gauthier-Villars, Paris, 1926.
I. Babuška and Manil Suri, The optimal convergence rate of the $p$-version of the finite element method, SIAM J. Numer. Anal. 24 (1987), no. 4, 750–776. MR 899702, DOI 10.1137/0724049
Dietrich Braess and Christoph Schwab, Approximation on simplices with respect to weighted Sobolev norms, J. Approx. Theory 103 (2000), no. 2, 329–337. MR 1749969, DOI 10.1006/jath.1999.3429
C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral methods, Scientific Computation, Springer-Verlag, Berlin, 2006. Fundamentals in single domains. MR 2223552
Moshe Dubiner, Spectral methods on triangles and other domains, J. Sci. Comput. 6 (1991), no. 4, 345–390. MR 1154903, DOI 10.1007/BF01060030
Charles F. Dunkl and Yuan Xu, Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, vol. 81, Cambridge University Press, Cambridge, 2001. MR 1827871, DOI 10.1017/CBO9780511565717
Daniele Funaro, Polynomial approximation of differential equations, Lecture Notes in Physics. New Series m: Monographs, vol. 8, Springer-Verlag, Berlin, 1992. MR 1176949
Axel Grundmann and H. M. Möller, Invariant integration formulas for the $n$-simplex by combinatorial methods, SIAM J. Numer. Anal. 15 (1978), no. 2, 282–290. MR 488881, DOI 10.1137/0715019
Ben-Yu Guo, Jie Shen, and Li-Lian Wang, Optimal spectral-Galerkin methods using generalized Jacobi polynomials, J. Sci. Comput. 27 (2006), no. 1-3, 305–322. MR 2285783, DOI 10.1007/s10915-005-9055-7
Ben-Yu Guo, Jie Shen, and Li-Lian Wang. Generalized Jacobi polynomials/functions and applications to spectral methods. Appl. Numer. Math., 59(5):1011–1028, 2009.
Ben-yu Guo and Li-Lian Wang, Error analysis of spectral method on a triangle, Adv. Comput. Math. 26 (2007), no. 4, 473–496. MR 2291668, DOI 10.1007/s10444-005-7471-8
Wilhelm Heinrichs, Spectral collocation on triangular elements, J. Comput. Phys. 145 (1998), no. 2, 743–757. MR 1645013, DOI 10.1006/jcph.1998.6052
J. S. Hesthaven and D. Gottlieb, Stable spectral methods for conservation laws on triangles with unstructured grids, Comput. Methods Appl. Mech. Engrg. 175 (1999), no. 3-4, 361–381. MR 1702193, DOI 10.1016/S0045-7825(98)00361-2
George Em Karniadakis and Spencer J. Sherwin, Spectral/$hp$ element methods for computational fluid dynamics, 2nd ed., Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2005. MR 2165335, DOI 10.1093/acprof:oso/9780198528692.001.0001
Tom Koornwinder, Two-variable analogues of the classical orthogonal polynomials, Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) Math. Res. Center, Univ. Wisconsin, Publ. No. 35, Academic Press, New York, 1975, pp. 435–495. MR 0402146
Heping Ma and Weiwei Sun, A Legendre-Petrov-Galerkin and Chebyshev collocation method for third-order differential equations, SIAM J. Numer. Anal. 38 (2000), no. 5, 1425–1438. MR 1812518, DOI 10.1137/S0036142999361505
Steven A. Orszag, Spectral methods for problems in complex geometries, J. Comput. Phys. 37 (1980), no. 1, 70–92. MR 584322, DOI 10.1016/0021-9991(80)90005-4
R. G. Owens, Spectral approximations on the triangle, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454 (1998), no. 1971, 857–872. MR 1631583, DOI 10.1098/rspa.1998.0189
Ch. Schwab, $p$- and $hp$-finite element methods, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 1998. Theory and applications in solid and fluid mechanics. MR 1695813
Jie Shen, Li-Lian Wang, and Huiyuan Li. A triangular spectral element method using fully tensorial rational basis functions. SIAM J. Numer. Anal., 47(3):1619–1650, 2009.
Spencer J. Sherwin and George Em. Karniadakis, A new triangular and tetrahedral basis for high-order $(hp)$ finite element methods, Internat. J. Numer. Methods Engrg. 38 (1995), no. 22, 3775–3802. MR 1362003, DOI 10.1002/nme.1620382204
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
A. H. Stroud, Integration formulas and orthogonal polynomials for two variables, SIAM J. Numer. Anal. 6 (1969), 222–229. MR 261788, DOI 10.1137/0706020