Abstract
We introduce the generalized Jacobi-Gauss-Lobatto interpolation involving the values of functions and their derivatives at the endpoints, which play important roles in the Jacobi pseudospectral methods for high order problems. We establish some results on these interpolations in non-uniformly weighted Sobolev spaces, which serve as the basic tools in analysis of numerical quadratures and various numerical methods of differential and integral equations.
Similar content being viewed by others
References
Askey R. Orthogonal Polynomials and Special Functions. Regional Conference Series in Applied Mathematics, Vol 21. Philadelphia: SIAM, 1975
Babuška I, Guo B Q. Direct and inverse approximation theorems for the p-version of finite element method in the framework of weighted Besov spaces. Part I: Approximability of functions in the weighted Besov space. SIAM J Numer Anal, 2001, 39: 1512–1538
Belhachmi Z, Bernardi C, Karageorghis A. Spectral element discretization of the circular driven cavity. Part II: The bilaplacian equation. SIAM J Numer Anal, 2001, 38: 1926–1960
Bergh J, Löfström J. Interpolation Spaces, An Introduction. Berlin: Spinger-Verlag, 1976
Bernardi C, Dauge M, Maday Y. Spectral Methods for Axisymmetric Domains. Series in Applied Mathematics, Vol 3. Paris: Gauhtier-Villars & North-Holland, 1999
Bernardi C, Maday Y. Spectral methods. In: Ciarlet P G, Lions P L, eds. Handbook of Numerical Analysis, Vol 5, Techniques of Scientific Computing. Amsterdam: Elsevier, 1997, 209–486
Bernardi C, Maday Y. Some spectral approximations of one-dimensional fourth-order problems. In: Nevai P, Pinkus A, eds. Progress in Approximation Theory. San Diego: Academic Press, 1991, 43–116
Bernardi C, Maday Y. Polynomial interpolation results in Sobolev spaces. J Comput Appl Math, 1992, 43: 53–80
Bialecki B, Karageorghis A. A Legendre spectral Galerkin method for the biharmonic Dirichlet problem. SIAM J Sci Comput, 2000, 22: 1549–1569
Bjørstad P E, Tjøstheim B P. Efficient algorithms for solving a fourth-order equation with spectral-Galerkin method. SIAM J Sci Comput, 1997, 18: 621–632
Boyd J P. Chebyshev and Fourier Spectral Methods. 2nd ed. Mineola: Dover, 2001
Canuto C, Hussaini M Y, Quarteroni A, Zang T A. Spectral Methods in Fluid Dynamics. Berlin: Springer-Verlag, 1998
Dubiner M. Spectral methods on triangles and other domains. J Sci Comput, 1991, 6: 345–390
Ezzirani A, Guessab A. A fast algorithm for Gaussian type quadrature formulae with mixed boundary conditions and some lumped mass spectral approximations. Math Comp, 1999, 225: 217–248
Gottlieb D, Orszag S A. Numerical Analysis of Spectral Methods: Theory and Applications. Philadelphia: SIAM, 1977
Guo B Y. Spectral Methods and Their Applications. Singapore: World Scientific, 1998
Guo B Y. Gegenbauer approximation and its applications to differential equations on the whole line. J Math Anal Appl, 1998, 226: 180–206
Guo B Y. Jacobi spectral approximation and its applications to differential equations on the half line. J Comput Math, 2000, 18: 95–112
Guo B Y. Gegenbauer approximation in certain Hilbert spaces and its applications to singular differential equations on the whole line. SIAM J Numer Anal, 2000, 37: 621–645
Guo B Y. Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations. J Math Anal Appl, 2000, 243: 373–408
Guo B Y, Shen J, Wang Z Q. A rational approximation and its applications to differential equations on the half line. J Sci Comput, 2000, 15: 117–148
Guo B Y, Shen J, Wang Z Q. Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval. Int J Numer Meth Engrg, 2002, 53: 65–84
Guo B Y, Wang L L. Jacobi interpolation approximations and their applications to singular differential equations. Adv Comput Math, 2000, 14: 227–276
Guo B Y, Wang L L. Error analysis of spectral method on a triangle. Adv Comput Math, 2007, 26: 473–496
Guo B Y, Wang L L. Jacobi approximations and Jacobi-Gauss-type interpolations in non-uniformly Jacobi-weighted Sobolev spaces. J Approx Theory, 2004, 28: 1–41
Guo B Y, Wang Z Q, Wan Z S, Chu D L. Second order Jacobi approximation with applications to fourth-order differential equations. Appl Numer Math, 2005, 55: 480–502
Hardy G H, Littlewood J E, Pólya G. Inequalities. Cambridge: Cambridge University Press, 1952
Junghanns V P. Uniform convergence of approximate methods for Cauchy type singular equation over (−1, 1). Wissenschaftliche Zeitschrift Technische Hocschule, Karl-Mars Stadt, 1984, 26: 250–256
Karniadakis G, Sherwin S. Spectral/hp Element Methods for Computational Fluid Dynamics. Oxford: Oxford University Press, 1999
Owens R G. Spectral approximation on the triangle. Proc R Soc Lond Ser A, 1998, 454: 857–872
Shen J. Efficient spectral-Galerkin method. I. Direct solvers of second- and fourth-order equations using Legendre polynomials. SIAM J Sci Comput, 1994, 15: 1489–1505
Sherwin S J, Farniadakis G E. A new triangular and tetrahedral basis for high-order finite element methods. Int J Numer Methods Engrg, 1995, 38: 3775–3802
Stephan E P, Suri M. On the convergence of the p-version of the boundary element Galerkin method. Math Comp, 1989, 52: 31–48
Szegö G. Orthogonal Polynomials. Providence: Amer Math Soc, 1959
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wan, Z., Guo, B. & Zhang, C. Generalized Jacobi-Gauss-Lobatto interpolation. Front. Math. China 8, 933–960 (2013). https://doi.org/10.1007/s11464-013-0271-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11464-013-0271-4
Keywords
- Generalized Jacobi-Gauss-Lobatto interpolation
- pseudospectral method
- non-uniformly weighted Sobolev space