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Optimal Error Estimates of the Legendre Tau Method for Second-Order Differential Equations

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Abstract

In this paper, we prove that the Legendre tau method has the optimal rate of convergence in L 2-norm, H 1-norm and H 2-norm for one-dimensional second-order steady differential equations with three kinds of boundary conditions and in C([0,T];L 2(I))-norm for the corresponding evolution equation with the Dirichlet boundary condition. For the generalized Burgers equation, we develop a Legendre tau-Chebyshev collocation method, which can also be optimally convergent in C([0,T];L 2(I))-norm. Finally, we give some numerical examples.

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References

  1. Aliabadi, M.H., Ortiz, E.L.: Numerical treatment of moving and free boundary value problems with the tau method. Comput. Math. Appl. 35, 53–61 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alpert, B.K., Rokhlin, V.: A fast algorithm for the evaluation of Legendre expansions. SIAM J. Sci. Stat. Comput. 12, 158–179 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  3. Babuška, I., Guo, B.Q.: Direct and inverse approximation theorems for the p-version of the finite element method in the framework of weighted Besov spaces I. Approximability of functions in the weighted Besov spaces. SIAM J. Numer. Anal. 39, 1512–1538 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bernardi, C., Maday, Y.: Spectral methods. In: Techniques of Scientific Computing (Part 2). Handbook of Numerical Analysis, vol. V, pp. 209–486. Elsevier, Amsterdam (1997)

    Google Scholar 

  5. Canuto, C., Hussaini, M.Y., Quartesoni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (1987)

    Google Scholar 

  6. Canuto, C., Hussaini, M.Y., Quartesoni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006)

    MATH  Google Scholar 

  7. Charalambides, M., Waleffe, F.: Spectrum of the Jacobi tau approximation for the second derivative operator. SIAM J. Numer. Anal. 46, 280–294 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dawkins, P.T., Dunbar, S.R., Douglass, R.W.: The origin and nature of spurious eigenvalues in the spectral tau method. J. Comput. Phys. 147, 441–462 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. Don, W.S., Gottlieb, D.: The Chebyshev-Legendre method: implementing Legendre methods on Chebyshev points. SIAM J. Numer. Anal. 31, 1519–1534 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  10. Elbarbary, E.: Efficient Chebyshev-Petrov-Galerkin method for solving second-order equations. J. Sci. Comput. 34, 113–126 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. El-Daou, M.K.: A posteriori error bounds for the approximate solution of second-order ODEs by piecewise coefficients perturbation methods. J. Comput. Appl. Math. 189, 51–66 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. El-Daou, M.K., Ortiz, E.L.: Error analysis of the tau method: dependence of the error on the degree and the length of the interval of approximation. Comput. Math. Appl. 25, 33–45 (1992)

    Article  Google Scholar 

  13. Funaro, D.: Polynomial Approximations of Differential Equations. Springer, Berlin (1992)

    Google Scholar 

  14. Guo, B.Y., Shen, J., Wang, L.L.: Optimal Spectral-Galerkin Methods using generalized Jacobi polynomials. J. Sci. Comput. 27, 305–322 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1952), pp. 240–244

    MATH  Google Scholar 

  16. Hosseini, S.M., Shahmorad, S.: Tau numerical solution of Fredholm integro-differential equations with arbitrary polynomial bases. Appl. Math. Model. 27, 145–154 (2003)

    Article  MATH  Google Scholar 

  17. Hosseini, S.M., Shahmorad, S.: Numerical piecewise approximate solution of Fredholm integro-differential equations by the Tau method. Appl. Math. Model. 29, 1005–1021 (2005)

    Article  MATH  Google Scholar 

  18. Li, H.Y.: Super-spectral Viscosity Methods for Nonlinear Conservation Laws, Chebyshev Collocation Methods and Their Applications. Shanghai University Press, Shanghai (2002)

    Google Scholar 

  19. Li, J., Ma, H.P., Sun, W.W.: Error analysis for solving the Korteweg-de Vries equation by a Legendre pseudospectral method. Numer. Methods Part D.E. 16, 513–534 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  20. Li, H.Y., Wu, H., Ma, H.P.: The Legendre Galerkin-Chebyshev collocation method for Burgers-like equations. IMA J. Numer. Anal. 23, 109–124 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ma, H.P.: Chebyshev-Legendre spectral viscosity method for nonlinear conservation laws. SIAM J. Numer. Anal. 35, 869–892 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  22. Ma, H.P., Sun, W.W.: A Legendre-Petrov-Galerkin and Chebyshev collocation method for third-order differential equations. SIAM J. Numer. Anal. 38, 1425–1438 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. Ma, H.P., Sun, W.W.: Optimal error estimates of the Legendre-Petrov-Galerkin method for the Korteweg-de Vries equation. SIAM J. Numer. Anal. 39, 1380–1394 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  24. Ma, H.P., Sun, W.W.: A coupled Legendre Petrov-Galerkin and collocation method for the generalized Korteweg-de Vries equations. In: Advances in Mathematics Research, vol. 2, pp. 53–73. Nova Science, Hauppauge (2003)

    Google Scholar 

  25. Sacchi Landriani, G.: Spectral tau approximation of the two-dimensional Stokes problem. Numer. Math. 52, 683–699 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  26. Shen, J.: A spectral-tau approximation for the Stokes and Navier-Stokes equations. Math. Model. Numer. Anal. 22, 677–693 (1988)

    MATH  Google Scholar 

  27. 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)

    Article  MATH  MathSciNet  Google Scholar 

  28. Shen, J.: Efficient spectral-Galerkin method II. Direct solvers of second- and fourth-order equations using Chebyshev polynomials. SIAM J. Sci. Comput. 16, 74–87 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  29. Shen, J.: Efficient Chebyshev-Legendre Galerkin methods for elliptic problems. In: Proceedings of ICOSAHOM’95, Houston J. Math., 233–239 (1996)

  30. Shen, J.: 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 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  31. Shen, J., Wang, L.L.: Legendre and Chebyshev dual-Petrov-Galerkin methods for hyperbolic equations. Comput. Methods Appl. Mech. Eng. 196, 3785–3797 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  32. Tang, J.G., Ma, H.P.: Single and multi-interval Legendre tau-methods in time for parabolic equations. Adv. Comput. Math. 17, 349–367 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  33. Wu, H., Ma, H.P., Li, H.Y.: Optimal error estimates of the Chebyshev-Legendre spectral method for solving the generalized Burgers equation. SIAM J. Numer. Anal. 41, 659–672 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  34. Yuan, J.M., Shen, J., Wu, J.: A dual-Petrov-Galerkin method for the Kawahara-type equations. J. Sci. Comput. 34, 48–63 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Ting-Ting Shen, Zhong-Qiang Zhang & He-Ping Ma

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  1. Ting-Ting Shen
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  2. Zhong-Qiang Zhang
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  3. He-Ping Ma
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Correspondence to He-Ping Ma.

Additional information

This work was supported by the NNSF of China (60874039), Shanghai Leading Academic Discipline Project (J50101), and Graduate Innovative Foundation of Shanghai University (shucx080129).

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Shen, TT., Zhang, ZQ. & Ma, HP. Optimal Error Estimates of the Legendre Tau Method for Second-Order Differential Equations. J Sci Comput 42, 198–215 (2010). https://doi.org/10.1007/s10915-009-9323-z

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  • DOI: https://doi.org/10.1007/s10915-009-9323-z

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