A Legendre Spectral Collocation Method for the Biharmonic Dirichlet Problem
Bernard Bialecki; Andreas Karageorghis
ESAIM: Mathematical Modelling and Numerical Analysis (2010)
- Volume: 34, Issue: 3, page 637-662
- ISSN: 0764-583X
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topBialecki, Bernard, and Karageorghis, Andreas. "A Legendre Spectral Collocation Method for the Biharmonic Dirichlet Problem." ESAIM: Mathematical Modelling and Numerical Analysis 34.3 (2010): 637-662. <http://eudml.org/doc/197522>.
@article{Bialecki2010,
abstract = {
A Legendre spectral collocation method is presented for the solution
of the biharmonic Dirichlet problem on a square. The solution and
its Laplacian are approximated using the set of basis functions suggested
by Shen, which are linear combinations of Legendre polynomials. A Schur
complement approach is used to reduce the resulting linear system to one
involving the approximation of the Laplacian of the solution on the two
vertical sides of the square. The Schur complement system is solved by
a preconditioned conjugate gradient method. The total cost of the algorithm
is O(N3). Numerical results demonstrate the spectral convergence of the method.
},
author = {Bialecki, Bernard, Karageorghis, Andreas},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Biharmonic Dirichlet problem; spectral collocation;
Schur complement; preconditioned conjugate gradient method.; biharmonic Dirichlet problem; Legendre spectral collocation method; Schur complement; preconditioned conjugate gradient method; biharmonic equation; numerical results},
language = {eng},
month = {3},
number = {3},
pages = {637-662},
publisher = {EDP Sciences},
title = {A Legendre Spectral Collocation Method for the Biharmonic Dirichlet Problem},
url = {http://eudml.org/doc/197522},
volume = {34},
year = {2010},
}
TY - JOUR
AU - Bialecki, Bernard
AU - Karageorghis, Andreas
TI - A Legendre Spectral Collocation Method for the Biharmonic Dirichlet Problem
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 3
SP - 637
EP - 662
AB -
A Legendre spectral collocation method is presented for the solution
of the biharmonic Dirichlet problem on a square. The solution and
its Laplacian are approximated using the set of basis functions suggested
by Shen, which are linear combinations of Legendre polynomials. A Schur
complement approach is used to reduce the resulting linear system to one
involving the approximation of the Laplacian of the solution on the two
vertical sides of the square. The Schur complement system is solved by
a preconditioned conjugate gradient method. The total cost of the algorithm
is O(N3). Numerical results demonstrate the spectral convergence of the method.
LA - eng
KW - Biharmonic Dirichlet problem; spectral collocation;
Schur complement; preconditioned conjugate gradient method.; biharmonic Dirichlet problem; Legendre spectral collocation method; Schur complement; preconditioned conjugate gradient method; biharmonic equation; numerical results
UR - http://eudml.org/doc/197522
ER -
References
top- C. Bernardi and Y. Maday, Spectral methods for the approximation of fourth order problems: Applications to the Stokes and Navier-Stokes equations. Comput. and Structures30 (1988) 205-216.
- 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.
- C. Bernardi, G. Coppoletta and Y. Maday, Some spectral approximations of two-dimensional fourth order problems. Math. Comp.59 (1992) 63-76.
- B. Bialecki, A fast solver for the orthogonal spline collocation solution of the biharmonic Dirichlet problem on rectangles. submitted.
- P.E. Bjørstad and B.P. Tjøstheim, Efficient algorithms for solving a fourth-order equation with the spectral-Galerkin method. SIAM J. Sci. Comput.18 (1997) 621-632.
- 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).
- W. Heinrichs, A stabilized treatment of the biharmonic operator with spectral methods. SIAM J. Sci. Stat. Comput.12 (1991) 1162-1172.
- A. Karageorghis, 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.
- A. Karageorghis, A fully conforming spectral collocation scheme for second and fourth order problems. Comput. Methods Appl. Mech. Engng.126 (1995) 305-314.
- A. Karageorghis and T.N. Phillips, Conforming Chebyshev spectral collocation methods for the solution of laminar flow in a constricted channel. IMA Journal Numer. Anal.11 (1991) 33-55.
- A. Karageorghis and T. Tang, A spectral domain decomposition approach for steady Navier-Stokes problems in circular geometries. Computers and Fluids25 (1996) 541-549.
- Z.-M. Lou, B. Bialecki, and G. Fairweather, Orthogonal spline collocation methods for biharmonic problems. Numer. Math.80 (1998) 267-303.
- W.W. Schultz, N.Y. Lee and J.P. Boyd, Chebyshev pseudospectral method of viscous flows with corner singularities. J. Sci. Comput.4 (1989) 1-19.
- J. Shen, Efficient spectral-Galerkin method I. Direct solvers of second- and forth-order equations using Legendre polynomials. SIAM J. Sci. Comput.15 (1994) 1489-1505.
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