Numerische Mathematik

, Volume 46, Issue 4, pp 505–520 | Cite as

Spectral methods for exterior elliptic problems

  • C. Canuto
  • S. I. Hariharan
  • L. Lustman
Article

Summary

This paper deals with spectral approximations for exterior elliptic problems in two dimensions. As in the conventional finite difference or finite element methods, it is found that the accuracy of the numerical solutions is limited by the order of the numerical farfield conditions. We introduce a spectral boundary treatment at infinity, which is compatible with the “infinite order” interior spectral scheme. Computational results are presented to demonstrate the spectral accuracy attainable. Although we deal with a simple Laplace problem throughout the paper, our analysis covers more complex and general cases.

Subject Classifications

AMS(MOS): 65 N 30 CR: G 1.8 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aziz, A.K., Dorr, M.R., Kellog, R.B.: A New Approximation Method for the Helmholtz Equation in an Exterior Domain. SIAM J. Numer. Anal.19, 899–908 (1982)CrossRefGoogle Scholar
  2. 2.
    Bayliss, A., Gunzburger, M.D., Turkel, E.: Boundary Conditions for the Numerical Solution of Elliptic Equations in Exterior Regions. SIAM J. Appl. Math.42, 430–451 (1982)CrossRefGoogle Scholar
  3. 3.
    Boyd, J.P.: The Optimization of Convergent for Chebyshev Polynomial Methods in Unbounded Domain. J. Comput. Phys.45, 43–79 (1982)CrossRefGoogle Scholar
  4. 4.
    Canuto, C., Quarteroni, A.: Preconditioned Minimal Residual Methods for Chebyshev Spectral Calculations. ICASE Report 83-28 (1983)Google Scholar
  5. 5.
    Funaro, D.: Analysis of the Du-Fort Frankel Method for Differential Systems. (To appear on RAIRO Numer. Anal.)Google Scholar
  6. 6.
    Fix, G.J., Marin, S.P.: Variational Methods in Underwater Acoustic Problems. J. Comput. Phys.28, 253–270 (1978)CrossRefGoogle Scholar
  7. 7.
    Goldstein, C.I.: The Finite Element Method with Non-uniform Mesh sizes Applied to the Exterior Helmholtz Problem. Numer. Math.38, 61–82 (1982)Google Scholar
  8. 8.
    Gottlieb, D., Hussaini, M.Y., Voigt, R.G.: Proceeding of the Workshop on Spectral Methods. (Hampton, Virginia, August 1982)-SIAM (Philadelphia) (1984)Google Scholar
  9. 9.
    Gottlieb, D., Lustman, L.: The Dufort-Frankel Chebyshev Method for Parabolic Initial Boundary Value Problems. Comput. Fluids, 107–120 (1983)Google Scholar
  10. 10.
    Gottlieb, D., Lustman, L.: The Spectrum of the Chebyshev Collocation Operator for the Heat-Equation. SIAM J. Numer. Anal.20, 909–921 (1983)Google Scholar
  11. 11.
    Grosch, C.E., Orszag, S.A.: Numerical Solution of Problems in Unbounded Regions: Coordinate transforms. J. Comput. Phys.25, 273–295 (1977)Google Scholar
  12. 12.
    Greenspan, D., Werner, P.: A Numerical Method for the Exterior Dirichlet Problem for the Reduced Wave Equation. Arch. Ration. Mech. Anal.23, 288–316 (1966)Google Scholar
  13. 13.
    Haldenwang, P., Labrosse, G., Abboudi, S., Deville, M.: Chebyshev 3-D Spectral and 2-D Pseudo Spectral Solvers for the Helmholtz Equation. J. Comput. Phys.55, 115–128 (1984)Google Scholar
  14. 14.
    Hsiao, G.C., MacCamy, R.C.: Solution of Boundary Value Problems by Integral Equations of the First Kind. SIAM Rev.15, 687–704 (1973)Google Scholar
  15. 15.
    Kriegsmann, G.A., Morawetz, C.S.: Solving the Helmholtz Equation for Exterior Problems with Variable Index of Refraction I: SIAM J. Sci. Stat. Comput.1, 371–385 (1980)Google Scholar
  16. 16.
    Marin, S.P.: A Finite Element Method for Problems Involving the Helmholtz Equation in Two-Dimensional Exterior Regions Ph. D. Thesis, Carnegie-Mellon University, Pittsburgh 1978Google Scholar
  17. 17.
    MacCamy, R.C., Marin, S. P.: A Finite Element Method for Exterior Interface Problems. Int. J. Math. Math. Sci.3, 311–350 (1980)Google Scholar
  18. 18.
    Morchoisne, Y.: Resolution of Navier-Stokes Equations by a Space-Time Spectral Method. Rech. Aerosp.5, 293–306 (1973)Google Scholar
  19. 19.
    Meijerink, J.A., Van der Vorst, H.A.: An Iterative Solution Method for Linear Systems of Which the Coefficient Matrix is a SymmetricM-Matrix. Math. Comput.31, 148–162 (1977)Google Scholar
  20. 20.
    Orszag, S.A.: Spectral Methods for Problems in Complex Geometries. J. Comput. Phys.37, 70–92 (1980)Google Scholar
  21. 21.
    Phillips, T.N., Zang, T.A., Hussaini, M.Y.: Preconditioners for the Spectral Multigrid Method. ICASE Report 83-48 (1983)Google Scholar
  22. 22.
    Young, D.M., Jea, K.C.: Generalized Conjugate Gradient Accelleration of Nonsymmetrizable Iterative Methods. Linear Algebra Appl.34, 159–194 (1980)Google Scholar
  23. 23.
    Wong, Y.S.: Numerical Methods in Thermal Problems. (R.W. Lewis, K. Morgan, eds.), pp. 967–979, Swansea: Pineridge Press 1979Google Scholar
  24. 24.
    Wong, Y.S., Hafez, M.M.: A Minimal Residual Method for Transonic Potential Flow. ICASE Report 82-15 (1982)Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • C. Canuto
    • 1
  • S. I. Hariharan
    • 2
    • 3
  • L. Lustman
    • 4
  1. 1.Instituto di Analisi Numerica del C.N.R.PaviaItaly
  2. 2.University of Tennessee Space InstituteTullahomaUSA
  3. 3.Institute for Computer Applications in Science and EngineeringNASA Langeley Research CenterHamptonUSA
  4. 4.Systems and Applied Sciences CorporationHamptonUSA

Personalised recommendations