Advances in Computational Mathematics

, Volume 32, Issue 4, pp 393–429 | Cite as

Composite Laguerre-Legendre spectral method for exterior problems



In this paper, we propose a composite Laguerre-Legendre spectral method for two-dimensional exterior problems. Results on the composite Laguerre-Legendre approximation, which is a set of piecewise mixed approximations coupled with domain decomposition, are established. These results play important roles in the related spectral methods for exterior problems. As examples of applications, the composite spectral schemes are provided for two model problems, with the convergence analysis. An efficient implementation is described. Numerical results demonstrate the spectral accuracy in space of this new approach, and confirm the analysis. The approximation results and techniques developed in this paper are also applicable to other problems defined on unbounded domains.


Composite Laguerre-Legendre spectral method Exterior problems 

Mathematics Subject Classifications (2000)

65M70 41A30 35J20 35K20 


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  1. 1.
    Bernardi, C., Maday, Y.: Spectral methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, pp. 209–486. Elsevier, Amsterdam (1997)Google Scholar
  2. 2.
    Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover, Mineda (2001)MATHGoogle Scholar
  3. 3.
    Coulaud, O., Funaro, D., Kavian, O.: Laguerre spectral approximation of elliptic problems in exterior domains. Comput. Methods Appl. Mech. Engrg. 80, 451–458 (1990)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods. Springer, Berlin (2006)MATHGoogle Scholar
  5. 5.
    Funaro, D.: Polynomial Approxiamtions of Differential Equations. Springer, Berlin (1992)Google Scholar
  6. 6.
    Funaro, D., Kavian, O.: Approximation of some diffusion evolution equation in unbounded domains by Hermite function. Math. Comp. 57, 597–619 (1999)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. SIAM-CBMS, Philadelphia (1977)MATHGoogle Scholar
  8. 8.
    Guo, B.-y.: Spectral Methods and their Applications. World Scientific, Singapore (1998)MATHGoogle Scholar
  9. 9.
    Guo, B.-y.: Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations. J. Math. Anal. Appl. 243, 373–408 (2000)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Guo, B.-y., Ma, H.-p.: Composite Legendre-Laguerre approximation in unbounded domains. J. Comput. Math. 19, 101–112 (2001)MATHMathSciNetGoogle Scholar
  11. 11.
    Guo, B.-y., Shen, J.: Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval. Numer. Math. 86, 635–654 (2000)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Guo, B.-y., Shen, J., Wang, L.-l.: Optimal spectral-Galerkin methods using generalized Jacobi polynomials. J. Sci. Comput. 27, 305–322 (2006)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Guo, B.-y., Shen, J., Xu, C.-L.: Generalized Laguerre approximation and its applications to exterion problems. J. Comput. Math. 23, 113–130 (2005)MATHMathSciNetGoogle Scholar
  14. 14.
    Guo, B.-y., Wang, L.-l.: Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces. J. Approx. Theory 128, 1–41 (2004)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Guo, B.-y., Wang, T.-j.: Composite generalized Laguerre-Legendre spectral method with its application to Fokker-Planck equation in an finite channel. Math. Comput. 78, 129–151 (2009)MathSciNetGoogle Scholar
  16. 16.
    Guo, B.-y., Xu, C.-L.: Mixed Laguerre-Legendre pseodospectral method for incompressible fluid flow in an infinite strip. Math. Comput. 72, 95–125 (2003)MathSciNetGoogle Scholar
  17. 17.
    Guo, B.-y., Zhang, X.-y.: A new generalized Laguerre approximation and its applications. J. Comput. Appl. Math. 181, 342–363 (2005)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Guo, B.-y., Zhang, X.-y.: Spectral method for differential equations of degenerate type by using generalized Laguerre functions. Appl. Numer. Math. 57, 455–471 (2007)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Maday, Y., Pernaud-Thomas, B., Vandeven, H.: Oneréhabilitation des méthods spèctrales de type Laguerre. Rech. Aérospat. 6 , 353–379 (1985) MathSciNetGoogle Scholar
  20. 20.
    Quarteroni, A.: Domain decomposition techniques using spectral methods. Calcolo 24, 141–177 (1987)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Quarteroni, A.: Domain decomposition methods for systems of conservation laws, spectral collocation approximations. SIAM J. Sci. Comput. 11, 1029–1052 (2004)MathSciNetGoogle Scholar
  22. 22.
    Shen, J.: Stable and efficient spectral methods in unbounded domains using Laguerre functions. SIAM J. Numer. Anal. 38, 1113–1133 (2000)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Wang, T.-j., Guo, B.-y.: Composite Laguerre-Legendre pseudospectral method for exterior problems. Commun. Comput. Phys. 5, 350–375 (2009)MathSciNetGoogle Scholar
  24. 24.
    Xu, C.-l., Guo, B.-y.: Mixed Laguerre-Legendre spectral method for incompressible fluid flow in an infinite strip. Adv. Comput. Math. 16, 77–96 (2002)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Xu, C.-l., Guo, B.-y.: Laguerre pseudospectral method for nonlinear partial differential equation. J. Comput. Math. 20, 413–428 (2002)MATHMathSciNetGoogle Scholar
  26. 26.
    Zhang, X.-y., Guo, B.-y.: Spherical harmonic-generalized Laguerre spectral method for exterior problems. J. Sci. Comput. 27, 305–322 (2006)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Normal UniversityShanghaiChina
  2. 2.Scientific Computing Key Laboratory of Shanghai UniversitiesShanghai UniversityShanghaiChina
  3. 3.Division of Computational Science of E-Institute of Shanghai UniversitiesShanghai UniversityShanghaiChina
  4. 4.Department of Mathematics and PhysicsHenan University of Science and TechnologyLuoYangChina

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