Abstract
In this paper, we investigate composite Laguerre-Legendre spectral method for fourth-order exterior problems. Some results on composite Laguerre-Legendre approximation are established, which is a set of piecewise mixed approximations coupled with domain decomposition. These results play an important role in spectral method for fourth-order exterior problems with rectangle obstacle. As examples of applications, composite spectral schemes are provided for two model problems, with convergence analysis. Efficient algorithms are implemented. Numerical results demonstrate their high accuracy, and confirm theoretical analysis well.
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References
Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods, Fundamentals in Single Domains. Springer, Berlin (2006)
Coulaud, O., Funaro, D., Kavian, O.: Laguerre spectral approximation of elliptic problems in exterior domains. Comput. Mech. Appl. Mech. Eng. 80, 451–458 (1990)
Funaro, D.: Polynomial Approximations of Differential Equations. Springer, Berlin (1992)
Funaro, D., Kavian, O.: Approximation of some diffusion evolution equation in unbounded domains by Hermite function. Math. Comput. 57, 597–619 (1999)
Guo, B.-Y.: Error estimation of Hermite spectral method for nonlinear partial differential equations. Math. Comput. 68, 1067–1078 (1999)
Guo, B.-Y., Jiao, Y.-J.: Mixed Generalized Laguerre-Fourier spectral method for exterior problem of Navier-Stokes equations. Discrete Continuous Dyn. Syst. B 11, 315–345 (2009)
Guo, B.-Y., Shen, J.: Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval. Numer. Math. 86, 635–654 (2000)
Guo, B.-Y., Wang, L.-L.: Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces. J. Approx. Theory 128, 1–41 (2004)
Guo, B.-Y., Wang, T.-J.: Composite generalized Laguerre-Legendre spectral method with domain decomposition and its application to Fokker-Planck equation in an finite channel. Math. Comput. 78, 129–151 (2009)
Guo, B.-Y., Wang, T.-J.: Composite Laguerre-Legendre spectral method for exterior problems. Adv. Comput. Math. 32, 393–429 (2010)
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)
Guo, B.-Y., Zhang, X.-Y.: A new generalized Laguerre approximation and its applications. J. Comput. Appl. Math. 181, 342–363 (2005)
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)
Guo, B.-Y., Shen, J., Xu, C.-L.: Generalized Laguerre approximation and its applications to exterior problems. J. Comput. Math. 23, 113–130 (2005)
Guo, B.-Y., Wang, Z.-Q., Wan, Z.-S., Chu, D.: Second order Jacobi approximations with applications to fourth-order differential equations. Appl. Num. Math. 55, 480–502 (2005)
Maday, Y., Pernaud-Thomas, B., Vandeven, H.: Oneréhabilitation des méthods spèctrales de type Laguerre. Rech. Aérosp. 6, 353–379 (1985)
Shen, J.: Efficient spectral-Galerkin method I, Direct solvers of second and fourth order equations using Legendre polynomials. SIAM J. Sci. Comput. 15, 1489–1505 (1994)
Shen, J.: Stable and efficient spectral methods in unbounded domains using Laguerre functions. SIAM J. Numer. Anal. 38, 1113–1133 (2000)
Shen, J., Wang, L.-L.: Some recent advances on spectral methods for unbounded domains. Commun. Comput. Phys. 5, 159–241 (2009)
Zhang, X.-Y., Guo, B.-Y.: Spherical harmonic-generalized Laguerre spectral method for exterior problems. J. Sci. Comput. 27, 305–322 (2006)
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The work of B.-Y. Guo is supported in part by NSF of China, N. 10871131, The Science and Technology Commission of Shanghai Municipality, Grant N. 75105118, The Shanghai Leading Academic Discipline Project N. S30405, and The Fund for E-institutes of Shanghai Universities N. E03004.
The work of T.-J. Wang is supported in part by NSF of China N. 10871131 and The Doctor Fund of Henan University of Science and Technology N. 09001263.
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Guo, BY., Wang, TJ. Composite Laguerre-Legendre Spectral Method for Fourth-Order Exterior Problems. J Sci Comput 44, 255–285 (2010). https://doi.org/10.1007/s10915-010-9367-0
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DOI: https://doi.org/10.1007/s10915-010-9367-0