Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-22T21:56:34.214Z Has data issue: false hasContentIssue false
  • English
  • Français

A Legendre Spectral Collocation Method for the Biharmonic Dirichlet Problem

Published online by Cambridge University Press:  15 April 2002

Bernard Bialecki  and
Andreas Karageorghis
Bernard Bialecki
Affiliation:
Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado 80401, U.S.A. ([email protected])
Andreas Karageorghis
Affiliation:
Department of Mathematics and Statistics, University of Cyprus, P.O. Box 537, 1678 Nicosia, Cyprus.
Get access

Abstract

A Legendre spectral collocation method is presented for the solutionof the biharmonic Dirichlet problem on a square. The solution andits Laplacian are approximated using the set of basis functions suggestedby Shen, which are linear combinations of Legendre polynomials. A Schurcomplement approach is used to reduce the resulting linear system to oneinvolving the approximation of the Laplacian of the solution on the twovertical sides of the square. The Schur complement system is solved bya preconditioned conjugate gradient method. The total cost of the algorithmis O(N 3). Numerical results demonstrate the spectral convergence of the method.

Keywords

Biharmonic Dirichlet problemspectral collocationSchur complementpreconditioned conjugate gradient method.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 3 , May 2000 , pp. 637 - 662
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bernardi, C. and Maday, Y., Spectral methods for the approximation of fourth order problems: Applications to the Stokes and Navier-Stokes equations. Comput. and Structures 30 (1988) 205-216. CrossRef
C. Bernardi and Y. Maday, Some spectral approximations of one-dimensional fourth order problems, in: Progress in Approximation Theory, P. Nevai and A. Pinkus Eds., Academic Press, San Diego (1991), 43-116.
C. Bernardi and Y. Maday, Spectral methods, in Handbook of Numerical Analysis, Vol. V, Part 2: Techniques of Scientific Computing, P.G. Ciarlet and J.L. Lions Eds., North-Holland, Amsterdam (1997) 209-485.
Bernardi, C., Coppoletta, G. and Maday, Y., Some spectral approximations of two-dimensional fourth order problems. Math. Comp. 59 (1992) 63-76. CrossRef
B. Bialecki, A fast solver for the orthogonal spline collocation solution of the biharmonic Dirichlet problem on rectangles. submitted.
Bjørstad, P.E. and Tjøstheim, B.P., Efficient algorithms for solving a fourth-order equation with the spectral-Galerkin method. SIAM J. Sci. Comput. 18 (1997) 621-632. CrossRef
J.P. Boyd, Chebyshev and Fourier Spectral Methods. Springer-Verlag, Berlin (1989).
J. Douglas Jr. and T. Dupont, Collocation Methods for Parabolic Equations in a Single Space Variable. Lect. Notes Math. 358, Springer-Verlag, New York, 1974.
G.H. Golub and C.F. van Loan, Matrix Computations, Third edn., The Johns Hopkins University Press, Baltimore, MD (1996).
Heinrichs, W., A stabilized treatment of the biharmonic operator with spectral methods. SIAM J. Sci. Stat. Comput. 12 (1991) 1162-1172. CrossRef
Karageorghis, A., The numerical solution of laminar flow in a re-entrant tube geometry by a Chebyshev spectral element collocation method. Comput. Methods Appl. Mech. Engng. 100 (1992) 339-358. CrossRef
Karageorghis, A., A fully conforming spectral collocation scheme for second and fourth order problems. Comput. Methods Appl. Mech. Engng. 126 (1995) 305-314. CrossRef
Karageorghis, A. and Phillips, T.N., Conforming Chebyshev spectral collocation methods for the solution of laminar flow in a constricted channel. IMA Journal Numer. Anal. 11 (1991) 33-55. CrossRef
Karageorghis, A. and Tang, T., A spectral domain decomposition approach for steady Navier-Stokes problems in circular geometries. Computers and Fluids 25 (1996) 541-549. CrossRef
Lou, Z.-M., Bialecki, B., and Fairweather, G., Orthogonal spline collocation methods for biharmonic problems. Numer. Math. 80 (1998) 267-303. CrossRef
Schultz, W.W., Lee, N.Y. and Boyd, J.P., Chebyshev pseudospectral method of viscous flows with corner singularities. J. Sci. Comput. 4 (1989) 1-19. CrossRef
Shen, J., Efficient spectral-Galerkin method I. Direct solvers of second- and forth-order equations using Legendre polynomials. SIAM J. Sci. Comput. 15 (1994) 1489-1505. CrossRef