Numerical Algorithms

, Volume 54, Issue 4, pp 533–569 | Cite as

Quasi-Lp norm orthogonal Galerkin expansions in sums of Jacobi polynomials

Orthogonal expansions
Open Access
Original Paper

Abstract

In the study of differential equations on [ − 1,1] subject to linear homogeneous boundary conditions of finite order, it is often expedient to represent the solution in a Galerkin expansion, that is, as a sum of basis functions, each of which satisfies the given boundary conditions. In order that the functions be maximally distinct, one can use the Gram-Schmidt method to generate a set orthogonal with respect to a particular weight function. Here we consider all such sets associated with the Jacobi weight function, w(x) = (1 − x)α(1 + x)β. However, this procedure is not only cumbersome for sets of large degree, but does not provide any intrinsic means to characterize the functions that result. We show here that each basis function can be written as the sum of a small number of Jacobi polynomials, whose coefficients are found by imposing the boundary conditions and orthogonality to the first few basis functions only. That orthogonality of the entire set follows—a property we term “auto-orthogonality”—is remarkable. Additionally, these basis functions are shown to behave asymptotically like individual Jacobi polynomials and share many of the latter’s useful properties. Of particular note is that these basis sets retain the exponential convergence characteristic of Jacobi expansions for expansion of an arbitrary function satisfying the boundary conditions imposed. Further, the associated error is asymptotically minimized in an Lp(α) norm given the appropriate choice of α = β. The rich algebraic structure underlying these properties remains partially obscured by the rather difficult form of the non-standard weighted integrals of Jacobi polynomials upon which our analysis rests. Nevertheless, we are able to prove most of these results in specific cases and certain of the results in the general case. However a proof that such expansions can satisfy linear boundary conditions of arbitrary order and form appears extremely difficult.

Keywords

Orthogonal polynomial Galerkin expansion Jacobi polynomial Spectral method Exponential convergence 

References

  1. 1.
    Askey, R.: Orthogonal Polynomials and Special Functions. SIAM, Philadelphia (1975)Google Scholar
  2. 2.
    Andrews, G.E., Burge, W.H.: Determinant identities. Pac. J. Math. 158, 1–14 (1993)MATHMathSciNetGoogle Scholar
  3. 3.
    Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Dover, New York (2001)MATHGoogle Scholar
  4. 4.
    Doha, E.H., Bhrawy, A.H.: A Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equations. Numer. Methods Partial Differ. Equ. 25(3) (2009)Google Scholar
  5. 5.
    Golub, G.H., Van Loan, C.F.: Matrix Computations, third edn. Johns Hopkins, Baltimore (1996)MATHGoogle Scholar
  6. 6.
    Gosper, R.W., Jr.: Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA 75, 40–42 (1977)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Ierley, G.R.: A class of sparse spectral operators for inversion of powers of the Laplacian in N dimensions. J. Sci. Comp. 12, 57–73 (1997)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ierley, G.R., Kerswell, R.R., Plasting, S.C.: Infinite Prandtl number convection. Part 2: a singular limit of upper bound theory. J. Fluid Mech. 560, 159–227 (2005)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Ismail, M.: Classical and quantum orthogonal polynomials in one variable. In: Encyclopedia of Mathematics and its Applications, vol. 98. CUP (2005)Google Scholar
  10. 10.
    Koekoek, R., Swarttouw, R.F.: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Delft University of Technology, Faculty of Information Technology and Systems, Department of Technical Mathematics and Informatics, Report no. 98-17 (1998)Google Scholar
  11. 11.
    Livermore, P.: Galerkin orthogonal polynomials. J. Comp. Phys. (2009, in press)Google Scholar
  12. 12.
    Livermore, P.: A compendium of Galerkin orthogonal polynomials. Tech. Rep., Scripps Institution of Oceanography, UC San Diego. Available at http://escholarship.org/uc/item/9vk1c6cm
  13. 13.
    Luke, Y.: The Special Functions and Their Approximations, vol. 1, first edn. Academic, London (1969)Google Scholar
  14. 14.
    Marcellan, F., Ronveaux, A.: On a class of polynomials orthogonal with respect to a discrete Sobolev inner product. Indag. Math., N.S. 1(4), 451–464 (1990)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Mason, J.C., Handscomb, D.C.: Chebyshev Polynomials. Chapman & Hall, London (2003)MATHGoogle Scholar
  16. 16.
    Szegö, G.: Orthogonal Polynomials, fourth edn. American Mathematical Society, Providence (1975)MATHGoogle Scholar
  17. 17.
    Weber, M., Erdelyi, A.: On the finite difference analogue of Rodrigues’ formula. Am. Math. Mon. 59, 163–168 (1952)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Institute of Geophysics and Planetary Physics, Scripps Institution of OceanographyUniversity of California, San DiegoLa JollaUSA
  2. 2.School of Earth and EnvironmentUniversity of LeedsLeedsUnited Kingdom

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