PRECONDITIONED SPECTRAL COLLOCATION METHOD ON CURVED ELEMENT DOMAINS USING THE GORDON-HALL TRANSFORMATION

Title & Authors
PRECONDITIONED SPECTRAL COLLOCATION METHOD ON CURVED ELEMENT DOMAINS USING THE GORDON-HALL TRANSFORMATION
Kim, Sang Dong; Hessari, Peyman; Shin, Byeong-Chun;

Abstract
The spectral collocation method for a second order elliptic boundary value problem on a domain $\small{{\Omega}}$ with curved boundaries is studied using the Gordon and Hall transformation which enables us to have a transformed elliptic problem and a square domain S
Keywords
spectral collocation method;Gordon and Hall transformation;elliptic equation;
Language
English
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References
1.
C. Canuto, P. Gervasio, and A. Quarteroni, Finite element preconditioning of G-NI spectral methods, SIAM J. Sci. Comput. 31 (2009/10), no. 6, 4422-4451.

2.
C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York, 1988.

3.
C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods, Fundamentals in Single Domains, Springer-Verlag, Berlin, 2006.

4.
C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods, Evolutions to Complex Geometries and Applications to Fluid Dynamics, Springer-Verlag, Berlin, 2007.

5.
M. O. Deville and E. H. Mund, Finite-element preconditioning for pseudospectral solutions of elliptic problems, SIAM J. Sci. Statist. Comput. 11 (1990), no. 2, 311-342.

6.
W. Fleming, Functions of Several Variables, Addison-Wesley, Reading, Mass., 1965.

7.
D. Funaro, Spectral elements for transport-dominated equations, Lecture notes in computational science and engineering, Vol. 1, Springer-Verlag, Berlin/Heidelberg, 1997.

8.
W. J. Gordon and C. A. Hall, Transfinite element methods: Blending function interpolation over arbitrary curved element domains, Numer. Math. 21 (1973), 109-129.

9.
W. J. Gordon and C. A. Hall, Geometric aspects of the finite element method: construction of curvilinear coordinate systems and their application to mesh generation, Int. J. Numer. Meth. Eng. 7 (1973), 461-477.

10.
W. Heinrichs, Spectral collocation scheme on the unit disc, J. Comput. Phys. 199 (2004), no. 1, 66-86.

11.
S. D. Kim and S. V. Parter, Preconditioning Chebyshev spectral collocation method for elliptic partial differential equations, SIAM J. Numer. Anal. 33 (1996), no. 6, 2375-2400.

12.
T. A. Manteuffel and J. Otto, Optimal equivalent preconditioners, SIAM J. Numer. Anal. 30 (1993), no. 3, 790-812.

13.
T. A. Manteuffel and S. V. Parter, Preconditioning and boundary conditions, SIAM J. Numer. Anal. 27 (1990), no. 3, 656-694.

14.
Y. Morochoisne, Resolution des equations de Navier-Stokes par une methode pseudo-spectrale en espace-temps, Rech. Aerospat. 1979 (1979), no. 5, 293-306.

15.
S. A. Orszag, Spectral methods for problems in complex geometries, J. Comput. Phys. 37 (1980), no. 1, 70-92.

16.
S. V. Parter and E. E. Rothman, Preconditioning Legendre spectral collocation approximation to elliptic problems, SIAM J. Numer. Anal. 32 (1995), no. 2, 333-385.

17.
S. V. Parter, Preconditioning Legendre spectral collocation methods for elliptic problems I. Finite differenc operators, SIAM J. Numer. Anal. 39 (2001), no. 1, 330-347.

18.
S. V. Parter, Preconditioning Legendre spectral collocation methods for elliptic problems II. Finite element operators, SIAM J. Numer. Anal. 39 (2001), no. 1, 348-362.

19.
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer-Verlag, Berlin, Heidelberg, 1994.

20.
W. R. Wade, An Introduction to Analysis, third edition, Prentice Hall, 2004.