Abstract
We extend the definition of the classical Jacobi polynomials withindexes α, β>−1 to allow α and/or β to be negative integers. We show that the generalized Jacobi polynomials, with indexes corresponding to the number of boundary conditions in a given partial differential equation, are the natural basis functions for the spectral approximation of this partial differential equation. Moreover, the use of generalized Jacobi polynomials leads to much simplified analysis, more precise error estimates and well conditioned algorithms.
Similar content being viewed by others
References
Babuška I., Suri M. (1987). The optimal convergence rate of the p-version of the finite element method. SIAM J. Numer. Anal. 24(4): 750–776
Babŭska I., Szabó B.A., Katz I.N. (1981). The p-version of the finite element method. SIAM J. Numer. Anal. 18: 512–545
Bernardi C., Maday Y. (1992). Approximations Spectrales de Problèmes aux Limites Elliptiques. Springer-Verlag, Paris
Bernardi C., Maday Y. (1997). Spectral method. In Ciarlet P. G., and Lions, L. L., (eds.), Handbook of Numerical Analysis, V. 5 (Part 2). North-Holland.
Bernardi, C., Dauge, M., and Maday, Y. (1999). Spectral Methods for Axisymmetric Domains, volume 3 of Series in Applied Mathematics (Paris). Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier, Paris. Numerical algorithms and tests due to Mejdi Azaï ez.
Bernardi C., Maday Y. (1991). Polynomial approximation of some singular functions. Appl. Anal. 42(1): 1–32
Dorr M.R. (1984). The approximation theory for the p-version of the finite element method. SIAM J. Numer. Anal. 21(6): 1180–1207
Funaro D. (1992). Polynomial Approxiamtions of Differential Equations. Springer-verlag.
Littlewood J.E., Hardy G.H., Pólya G. (1952). Inequalities. Cambridge University Press, UK
Guo B.Y. (2000). Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations. J. Math. Anal. Appl. 243: 373–408
Guo, B., Shen, J., and Wang, L.-L. Generalized Jacobi polynomials/functions and applications to spectral methods. Preprint.
Huang W.Z., Sloan D.M. (1992). The pseudospectral method for third-order differential equations. SIAM J. Numer. Anal. 29(6): 1626–1647
Merryfield W.J., Shizgal B. (1993). Properties of collocation third-derivative operators. J. Comput. Phys. 105(1): 182–185
Orszag S.A. (1980). Spectral methods for complex geometries. J. Comput. Phys. 37: 70–92
Parkes E.J., Zhu Z., Duffy B.R., Huang H.C. (1998). Sech-polynomial travelling solitary-wave solutions of odd-order generalized KdV-type equations. Physics Letters A 248: 219–224
Shen J. (1994). Efficient spectral-Galerkin method I. direct solvers for second- and fourth-order equations by using Legendre polynomials. SIAM J. Sci. Comput. 15: 1489–1505
Shen J. (2003). A new dual-Petrov-Galerkin method for third and higher odd-order differential equations: application to the KDV equation. SIAM J. Numer. Anal. 41: 1595–1619
Szegö, G. (1975). Orthogonal Polynomials (fourth edition). Volume 23. AMS Coll. Publications.
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics subject classification 1991. 65N35, 65N22, 65F05, 35J05
Rights and permissions
About this article
Cite this article
Guo, BY., Shen, J. & Wang, LL. Optimal Spectral-Galerkin Methods Using Generalized Jacobi Polynomials. J Sci Comput 27, 305–322 (2006). https://doi.org/10.1007/s10915-005-9055-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-005-9055-7