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Error Analysis for Mapped Jacobi Spectral Methods

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Abstract

Approximation properties of mapped Jacobi polynomials and of interpolations based on mapped Jacobi–Gauss–Lobatto points are established. These results play an important role in numerical analysis of mapped Jacobi spectral methods. As examples of applications, optimal error estimates for several popular regular and singular mappings are derived.

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References

  • R. Askey (1975) Orthogonal Polynomials and Special Functions Society for Industrial and Applied Mathematics Philadelphia, Pa

    Google Scholar 

  • Babuška. Ivo Guo. Benqi (2000) ArticleTitleOptimal estimates for lower and upper bounds of approximation errors in the p-version of the finite element method in two dimensions Numer. Math. 85 IssueID2 219–255

    Google Scholar 

  • A. Bayliss D. Gottlieb B.J. Matkowsky M. Minkoff (1989) ArticleTitleAn adaptive pseudo spectral method for reaction diffusion problems J. Comput. Phys. 81 421–443 Occurrence Handle10.1016/0021-9991(89)90215-5

    Article  Google Scholar 

  • A. Bayliss E. Turkel (1992) ArticleTitleMappings and accuracy for Chebyshev pseudo-spectral approximations J. Comput Phys. 101 IssueID2 349–359 Occurrence Handle10.1016/0021-9991(92)90012-N Occurrence HandleMR1174627

    Article  MathSciNet  Google Scholar 

  • J.P. Boyd (1987) ArticleTitleOrthogonal rational functions on a semi-infinite interval J. Comput. Phys. 70 63–88 Occurrence Handle10.1016/0021-9991(87)90002-7

    Article  Google Scholar 

  • J.P. Boyd (2001) Chebyshev and Fourier Spectral Methods Dover Publications Inc. Mineola, NY

    Google Scholar 

  • Canuto C., Hussaini M.Y., Quarteroni A., Zang T.A (1987). Spectral Methods in Fluid Dynamics, Springer-Verlag.

  • Courant R., Hilbert D. (1953). Methods of Mathematical Physics, Vol.1. Interscience Publishers.

  • R.A. Fisher (1937) ArticleTitleThe wave of advance of advantageous genes Ann. Eugenics. 7 335

    Google Scholar 

  • Funaro, D. (1992). Polynomial Approxiamtions of Differential Equations, Springer-verlag.

  • D. Gottlieb S.A. Orszag (1977) Numerical Analysis of Spectral Methods: Theory and Applications SIAM-CBMS Philadelphia

    Google Scholar 

  • Guo B.Y., Shen J., Wang L.L. Optimal spectral-Galerkin methods using generalized Jacobi polynomials. J. Sci. Comp. to appear.

  • B.Y. Guo L.L. Wang (2001) ArticleTitleJacobi interpolation approximations and their applications to singular differential equations Adv. Comput. Math. 14 227–276 Occurrence Handle10.1023/A:1016681018268

    Article  Google Scholar 

  • Guo, B. Y., Shen, J., and Wang, Z. Q. (2002). Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval Internat. J. Numer. Methods Engrg. 53 (1), 65–84 p and hp finite element methods, mathematics and engineering practice (St. Louis, MO, 2000).

    Google Scholar 

  • B.Y. Guo (2000) ArticleTitleJacobi approximations in certain Hilbert spaces and their applications to singular differential equations J Math. Anal. Appl. 243 373–408 Occurrence Handle10.1006/jmaa.1999.6677

    Article  Google Scholar 

  • B.Y. Guo (2000) ArticleTitleJacobi spectral approximation and its applications to differential equations on the half line J. Comput. Math. 18 95–112

    Google Scholar 

  • B.Y. Guo J. Shen Z.Q. Wang (2000) ArticleTitleA rational approximation and its applications to differential equations on the half line J. Sci. Comp. 15 117–147 Occurrence Handle10.1023/A:1007698525506

    Article  Google Scholar 

  • B. Guo L.L. Wang (2004) ArticleTitleJacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces J. Approx. theor. 128 1–41 Occurrence Handle10.1016/j.jat.2004.03.008

    Article  Google Scholar 

  • R.B. Kellogg A. Tsan (1978) ArticleTitleAnalysis of some difference approximations for a singular pertubation problem without turning points Math. Comp. 32 1025–1039

    Google Scholar 

  • D. Kosloff H. Tal-Ezer (1993) ArticleTitleA modified Chebyshev pseudospectral method with an o(n−1) time step restriction J. Comput. Phys. 104 457–469 Occurrence Handle10.1006/jcph.1993.1044

    Article  Google Scholar 

  • W.B. Liu J. Shen (1996) ArticleTitleA new efficient spectral- Galerkin method for singular perturbation problems J. Sci. Comput. 11 411–437 Occurrence Handle10.1007/BF02088955

    Article  Google Scholar 

  • W.B. Liu T. Tang (2001) ArticleTitleError analysis for a Galerkin-spectral method with coordinate transformation for solving singularly perturbed problems Appl. Numer. Math. 38 315–345 Occurrence Handle10.1016/S0168-9274(01)00036-8

    Article  Google Scholar 

  • L.S. Mulholland W.Z. Huang D.M. Sloan (1998) ArticleTitlePseudospectral solutions of near-singular problems using numerical coordinate transformations based on adaptivity SIAM J. Sci Comput. 19 1261–1289 Occurrence Handle10.1137/S1064827595291984

    Article  Google Scholar 

  • S.A. Orszag M. Israeli (1974) ArticleTitleNumerical simulation of viscous incompressible flows Ann. Rev. Fluid. Mech. 5 281–317 Occurrence Handle10.1146/annurev.fl.06.010174.001433

    Article  Google Scholar 

  • J. Shen L.L. Wang (2004) ArticleTitleError analysis for mapped Legendre spectral and pseudospectral methods SIAM J Numer. Anal. 42 326–349 Occurrence Handle10.1137/S0036142903422065

    Article  Google Scholar 

  • Y.H. Song D. Haidvogel (1994) ArticleTitleA semi-implicit ocean circulation model using a generalized topography-following coordinate system J. Comput. Phys. 115 228–244 Occurrence Handle10.1006/jcph.1994.1189

    Article  Google Scholar 

  • Szegö G. (1975). Orthogonal Polynomials, (4th ed.), Vol. 23, AMS Coll. Publ.

  • Tang, T., and Trummer, M. R. (1996). Boundary layer resolving pseudospectral mathod for singular pertubation problems. SIAM J. Sci. Comput. 17, 430–438.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Li-Lian Wang

  2. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, P.R. China

    Li-Lian Wang

  3. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Jie Shen

  4. Department of Mathematics, Xiamen University, Xiamen, China

    Jie Shen

Authors
  1. Li-Lian Wang
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  2. Jie Shen
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Correspondence to Jie Shen.

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Mathematics Subject Classification (1991): 65N35; 65N15; 65N50

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Wang, LL., Shen, J. Error Analysis for Mapped Jacobi Spectral Methods. J Sci Comput 24, 183–218 (2005). https://doi.org/10.1007/s10915-004-4613-y

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  • DOI: https://doi.org/10.1007/s10915-004-4613-y

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