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Optimal error estimates of the Legendre tau method for second-order differential equations. (English) Zbl 1203.65118
Summary: 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.

MSC:
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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[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) · Zbl 0999.65110 · doi:10.1016/S0898-1221(98)00044-3
[2] Alpert, B.K., Rokhlin, V.: A fast algorithm for the evaluation of Legendre expansions. SIAM J. Sci. Stat. Comput. 12, 158–179 (1991) · Zbl 0726.65018 · doi:10.1137/0912009
[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) · Zbl 1008.65078
[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)
[5] Canuto, C., Hussaini, M.Y., Quartesoni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (1987)
[6] Canuto, C., Hussaini, M.Y., Quartesoni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006) · Zbl 1093.76002
[7] Charalambides, M., Waleffe, F.: Spectrum of the Jacobi tau approximation for the second derivative operator. SIAM J. Numer. Anal. 46, 280–294 (2008) · Zbl 1160.65326 · doi:10.1137/060665907
[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) · Zbl 0924.65077 · doi:10.1006/jcph.1998.6095
[9] Don, W.S., Gottlieb, D.: The Chebyshev-Legendre method: implementing Legendre methods on Chebyshev points. SIAM J. Numer. Anal. 31, 1519–1534 (1994) · Zbl 0815.65106 · doi:10.1137/0731079
[10] Elbarbary, E.: Efficient Chebyshev-Petrov-Galerkin method for solving second-order equations. J. Sci. Comput. 34, 113–126 (2008) · Zbl 1136.65106 · doi:10.1007/s10915-007-9161-9
[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) · Zbl 1104.65084 · doi:10.1016/j.cam.2005.01.006
[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) · Zbl 0772.65054 · doi:10.1016/0898-1221(93)90310-R
[13] Funaro, D.: Polynomial Approximations of Differential Equations. Springer, Berlin (1992) · Zbl 0774.41010
[14] Guo, B.Y., Shen, J., Wang, L.L.: Optimal Spectral-Galerkin Methods using generalized Jacobi polynomials. J. Sci. Comput. 27, 305–322 (2006) · Zbl 1102.76047 · doi:10.1007/s10915-005-9055-7
[15] Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1952), pp. 240–244
[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) · Zbl 1047.65114 · doi:10.1016/S0307-904X(02)00099-9
[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) · Zbl 1099.65136 · doi:10.1016/j.apm.2005.02.003
[18] Li, H.Y.: Super-spectral Viscosity Methods for Nonlinear Conservation Laws, Chebyshev Collocation Methods and Their Applications. Shanghai University Press, Shanghai (2002)
[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) · Zbl 0964.65100 · doi:10.1002/1098-2426(200011)16:6<513::AID-NUM2>3.0.CO;2-#
[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) · Zbl 1020.65072 · doi:10.1093/imanum/23.1.109
[21] Ma, H.P.: Chebyshev-Legendre spectral viscosity method for nonlinear conservation laws. SIAM J. Numer. Anal. 35, 869–892 (1998) · Zbl 0912.35104 · doi:10.1137/S0036142995293900
[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) · Zbl 0986.65095 · doi:10.1137/S0036142999361505
[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) · Zbl 1008.65070 · doi:10.1137/S0036142900378327
[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) · Zbl 1069.65108
[25] Sacchi Landriani, G.: Spectral tau approximation of the two-dimensional Stokes problem. Numer. Math. 52, 683–699 (1988) · Zbl 0629.76037 · doi:10.1007/BF01395818
[26] Shen, J.: A spectral-tau approximation for the Stokes and Navier-Stokes equations. Math. Model. Numer. Anal. 22, 677–693 (1988) · Zbl 0657.76031
[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) · Zbl 0811.65097 · doi:10.1137/0915089
[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) · Zbl 0840.65113 · doi:10.1137/0916006
[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) · Zbl 1053.65085 · doi:10.1137/S0036142902410271
[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) · Zbl 1173.65342 · doi:10.1016/j.cma.2006.10.031
[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) · Zbl 1002.65111 · doi:10.1023/A:1016273820035
[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) · Zbl 1050.65083 · doi:10.1137/S0036142901399781
[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) · Zbl 1133.76040 · doi:10.1007/s10915-007-9158-4
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