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Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials. (English) Zbl 1103.65119

The authors are concerned with the efficiency of spectral-Galerkin methods in solving second order boundary value problems. Specifically, they use the Jacobi polynomials in order to construct shape functions bases. These bases satisfy the boundary conditions and, at the same time, lower the condition number of the accompanying matrices with two orders. Some numerical examples are carried out to underline the reliability of the chosen bases.

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65F05 Direct numerical methods for linear systems and matrix inversion
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65F35 Numerical computation of matrix norms, conditioning, scaling
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[1] Auteri, F., Quartapelle, L.: Galerkin spectral method for the vorticity and stream function equations. J. Comput. Phys. 149, 306–332 (1999) · Zbl 0934.76065 · doi:10.1006/jcph.1998.6155
[2] Auteri, F., Quartapelle, L.: Galerkin–Legendre spectral method for the 3D Helmholtz equation. J. Comput. Phys. 161, 454–483 (2000) · Zbl 0959.65130 · doi:10.1006/jcph.2000.6504
[3] Auteri, F., Parolini, N., Quartapelle, L.: Essential imposition of Neumann Galerkin–Legendre elliptic solvers. J. Comput. Phys. 185, 427–444 (2003) · Zbl 1017.65093 · doi:10.1016/S0021-9991(02)00064-5
[4] Ben-Yu, G.: Jacobi spectral approximations to differential equations on the half line. J. Comput. Math. 18, 95–112 (2000) · Zbl 0948.65071
[5] Ben-Yu, G.: Jacobi spectral method for differential equations with rough asymptotic behaviors at infinity. Comput. Math. Appl. 46, 95–104 (2003) · Zbl 1055.65096 · doi:10.1016/S0898-1221(03)90083-6
[6] Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd ed. Dover, Mineola (2001) · Zbl 0994.65128
[7] Buzbee, B.L., Golub, G.H., Neilson, C.W.: On direct methods for solving Poisson’s equations. SIAM J. Numer. Anal. 7, 627–656 (1970) · Zbl 0217.52902 · doi:10.1137/0707049
[8] Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, New York (1988) · Zbl 0658.76001
[9] Dang-Vu, H., Delcarte, C.: An accurate solution of the Poisson equation by the Chebyshev collocation method. J. Comput. Phys. 104, 211–220 (1993) · Zbl 0765.65107 · doi:10.1006/jcph.1993.1021
[10] Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration. Academic, New York (1975) · Zbl 0304.65016
[11] Doha, E.H.: An accurate double Chebyshev spectral approximation for Poisson’s equation. Ann. Univ. Sci. Budapest Sect. Comp. 10, 243–276 (1990) · Zbl 0717.65090
[12] Doha, E.H.: An accurate solution of parabolic equations by expansion in ultraspherical polynomials. J. Comput. Math. Appl. 19, 75–88 (1990) · Zbl 0706.65089 · doi:10.1016/0898-1221(90)90139-B
[13] Doha, E.H.: The coefficients of differentiated expansions and derivatives of ultraspherical polynomials. J. Comput. Math. Appl. 21, 115–122 (1991) · Zbl 0723.33008 · doi:10.1016/0898-1221(91)90089-M
[14] Doha, E.H.: The ultraspherical coefficients of the moments of a general-order derivative of an infinitely differentiable function. J. Comput. Appl. Math. 89, 53–72 (1998) · Zbl 0909.33007 · doi:10.1016/S0377-0427(97)00228-8
[15] Doha, E.H.: On the coefficients of differentiated expansions and derivatives of Jacobi polynomials. J. Phys. A: Math. Gen. 35, 3467–3478 (2002) · Zbl 0997.33004 · doi:10.1088/0305-4470/35/15/308
[16] Doha, E.H.: On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials. J. Phys. A: Math. Gen. 37, 657–675 (2004) · Zbl 1055.33007 · doi:10.1088/0305-4470/37/3/010
[17] Doha, E.H., Abd-Elhameed, W.M.: Efficient spectral-Galerkin algorithms for direct solution of second-order equations using ultraspherical polynomials. SIAM J. Sci. Comput. 24, 548–571 (2002) · Zbl 1020.65088 · doi:10.1137/S1064827500378933
[18] Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia, Pennsylvania (1977) · Zbl 0412.65058
[19] Haidvogel, D.B., Zang, T.: The accurate solution of Poisson’s equation by expansion in Chebyshev polynomials. J. Comput. Phys. 30, 167–180 (1979) · Zbl 0397.65077 · doi:10.1016/0021-9991(79)90097-4
[20] Heinrichs, W.: Improved condition number of spectral methods. Math. Comp. 53, 103–119 (1989) · Zbl 0676.65115 · doi:10.1090/S0025-5718-1989-0972370-0
[21] Heinrichs, W.: Spectral methods with sparse matrices. Numer. Math. 56, 25–41 (1989) · Zbl 0668.65089 · doi:10.1007/BF01395776
[22] Heinrichs, W.: Algebraic spectral multigrid methods. Comput. Methods Appl. Mech. Eng. 80, 281–286 (1990) · Zbl 0742.65085 · doi:10.1016/0045-7825(90)90031-G
[23] Lanczos, C.: Applied Analysis. Pitman, London (1957)
[24] Luke, Y.: The Special Functions and Their Approximations, vol. 1. Academic, New York (1969) · Zbl 0193.01701
[25] Orszag, S.A.: Spectral methods for problems in complex geometrics. J. Comput. Phys. 37, 70–92 (1980) · Zbl 0476.65078 · doi:10.1016/0021-9991(80)90005-4
[26] Ralston, A.: A First Course in Numerical Analysis. McGraw-Hill, New York (1965) · Zbl 0139.31603
[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] Siyyam, H.I., Syam, M.I.: An accurate solution of Poisson equation by the Chebyshev–Tau method. J. Comput. Appl. Math. 85, 1–10 (1997) · Zbl 0890.65115 · doi:10.1016/S0377-0427(97)00104-0
[30] Szegö, G.: Orthogonal Polynomials. Am. Math. Soc. Colloq. Pub. 23 (1985) · JFM 65.0278.03
[31] Watson, G.N.: A note on generalized hypergeometric series. Proc. Lond. Math. Soc. 23(2), xiii–xv (1925) (Records for 8 Nov. 1923) · JFM 51.0283.04
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