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Three-point finite-difference schemes, Padé and the spectral Galerkin method. I. One-sided impedance approximation


Authors: Vladimir Druskin and Shari Moskow
Journal: Math. Comp. 71 (2002), 995-1019
MSC (2000): Primary 65N06, 65N35
DOI: https://doi.org/10.1090/S0025-5718-01-01349-7
Published electronically: November 19, 2001
MathSciNet review: 1898743
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Abstract: A method for calculating special grid placement for three-point schemes which yields exponential superconvergence of the Neumann to Dirichlet map has been suggested earlier. Here we show that such a grid placement can yield impedance which is equivalent to that of a spectral Galerkin method, or more generally to that of a spectral Galerkin-Petrov method. In fact we show that for every stable Galerkin-Petrov method there is a three-point scheme which yields the same solution at the boundary. We discuss the application of this result to partial differential equations and give numerical examples. We also show equivalence at one corner of a two-dimensional optimal grid with a spectral Galerkin method.


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  • 1. N. I. Akhiezer, Theory of approximation, Dover, N.Y., 1992. MR 94b:01041
  • 2. S. Asvadurov, V. Druskin, and L. Knizhnerman, Application of the difference Gaussian rules to solution of hyperbolic problems, J. Comp. Phys, 158, 116-135 (2000). MR 2000j:76109
  • 3. G. A. Baker and P. Graves-Morris, Padé Approximants, Addison-Wesley Publishing Co., London et al., 1996. MR 97h:41001
  • 4. K. Black, Spectral elements on infinite domains, SIAM J. Sci. Comput., (1998) 19, 5, pp. 1667-1681. CMP 98:11
  • 5. J. P. Boyd, Chebyshev & Fourier spectral methods, Springer-Verlag, 1989.
  • 6. E. Braverman, M. Israeli, A. Averbuch, A fast spectral solver for a 3D Helmholtz equation, SIAM J. Sci. Comput. 1999, 20, No 6, pp. 2237-2260. MR 2000d:65201
  • 7. V. Druskin, Spectrally optimal finite-difference grids in unbounded domains, Schlumberger-Doll Research, Research Note, EMG-002-97-22 (1997).
  • 8. V. Druskin and L. Knizhnerman, Gaussian spectral rules for the three-point second differences: I. A two-point positive definite problem in a semiinfinite domain, SIAM J. Numer. Anal. 37, No 2, (1999), 403-422. MR 2000i:65163
  • 9. V. Druskin and L. Knizhnerman, Gaussian spectral rules for second order finite-difference schemes, Mathematical journey through analysis, matrix theory, and scientific computation. Numer. Algorithms 25 (2000), 139-159. CMP 2001:11
  • 10. V. Druskin and S. Moskow, Three-point finite difference schemes, Padé and the spectral Galerkin method: II. Multidomain two-dimensional impedance approximation, in preparation.
  • 11. B. Fornberg Practical guide to pseudospectral methods, Cambridge University Press, Cambridge, UK, 1996. MR 97g:65001
  • 12. A. A. Gonchar, E. A. Rakhmanov, and S. P. Suetin, On the rate of convergence of Padé approximants of orthogonal expansions, in Progress in Approximation Theory, A. A. Gonchar and E. B. Saff, eds., Springer-Verlag, 1992, pp. 169-190. MR 94h:41033
  • 13. A. A. Gonchar, G. L. Lopes, Markov's theorem for multipoint Padé approximants, Mat. SSSR Sbornik, 105(147) (1978), 512-524. 639. MR 81j:41027
  • 14. D. Ingerman, V. Druskin and L. Knizhnerman, Optimal finite-difference grids and rational approximations of square root. I. Elliptic problems, Comm. Pure. Appl. Math., 53, (2000), no. 8, pp. 1039-1066. MR 2001d:65140
  • 15. I. S. Kac and M. G. Krein, On the spectral functions of the string, Amer. Math. Soc. Transl., 103 (1974), pp. 19-102. MR 48:6969
  • 16. S. A. Orszag, Spectral methods for problems in complex geometries, J. Comp. Phys., 37 (1980), pp. 70-92. MR 83e:65188
  • 17. B. N. Parlett, The symmetric eigenvalue problem, Prentice-Hall, SIAM, 1998. MR 99c:65072
  • 18. P. Petrushev, V. Popov, Rational approximation of real functions, Encyclopedia of mathematics and its applications, 28, Cambridge Univ. Press, 1987. MR 89i:41022
  • 19. B. Szabó, I. Babuska, Finite Element Analysis, Wiley & Sons, 1991. MR 93f:73001

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Additional Information

Vladimir Druskin
Affiliation: Schlumberger-Doll Research, Old Quarry Rd, Ridgefield, Connecticut 06877
Email: druskin@ridgefield.sdr.slb.com

Shari Moskow
Affiliation: Department of Mathematics, University of Florida, 358 Little Hall, P.O. Box 118105, Gainesville, Florida 32611-8105
Email: moskow@math.ufl.edu

DOI: https://doi.org/10.1090/S0025-5718-01-01349-7
Keywords: Second order scheme, exponential superconvergence, pseudospectral, Galerkin-Petrov, rational approximations
Received by editor(s): December 2, 1999
Received by editor(s) in revised form: July 12, 2000, and September 26, 2000
Published electronically: November 19, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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