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http://dx.doi.org/10.4134/BKMS.2014.51.2.595

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

Kim, Sang Dong (Department of Mathematics Kyungpook National University)
Hessari, Peyman (Institute of Mechanical Engineering Technology Kyungpook National University)
Shin, Byeong-Chun (Department of Mathematics Chonnam National University)
Publication Information
Bulletin of the Korean Mathematical Society / v.51, no.2, 2014 , pp. 595-612 More about this Journal
Abstract
The spectral collocation method for a second order elliptic boundary value problem on a domain ${\Omega}$ 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 = [0, h] ${\times}$ [0, h], h > 0. The preconditioned system of the spectral collocation approximation based on Legendre-Gauss-Lobatto points by the matrix based on piecewise bilinear finite element discretizations is shown to have the high order accuracy of convergence and the efficiency of the finite element preconditioner.
Keywords
spectral collocation method; Gordon and Hall transformation; elliptic equation;
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1 D. Funaro, Spectral elements for transport-dominated equations, Lecture notes in computational science and engineering, Vol. 1, Springer-Verlag, Berlin/Heidelberg, 1997.
2 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.
3 C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York, 1988.
4 C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods, Fundamentals in Single Domains, Springer-Verlag, Berlin, 2006.
5 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.
6 W. Fleming, Functions of Several Variables, Addison-Wesley, Reading, Mass., 1965.
7 W. J. Gordon and C. A. Hall, Transfinite element methods: Blending function interpolation over arbitrary curved element domains, Numer. Math. 21 (1973), 109-129.   DOI
8 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.   DOI
9 W. Heinrichs, Spectral collocation scheme on the unit disc, J. Comput. Phys. 199 (2004), no. 1, 66-86.   DOI   ScienceOn
10 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.   DOI   ScienceOn
11 T. A. Manteuffel and J. Otto, Optimal equivalent preconditioners, SIAM J. Numer. Anal. 30 (1993), no. 3, 790-812.   DOI   ScienceOn
12 T. A. Manteuffel and S. V. Parter, Preconditioning and boundary conditions, SIAM J. Numer. Anal. 27 (1990), no. 3, 656-694.   DOI   ScienceOn
13 Y. Morochoisne, Resolution des equations de Navier-Stokes par une methode pseudo-spectrale en espace-temps, Rech. Aerospat. 1979 (1979), no. 5, 293-306.
14 S. A. Orszag, Spectral methods for problems in complex geometries, J. Comput. Phys. 37 (1980), no. 1, 70-92.   DOI   ScienceOn
15 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.   DOI   ScienceOn
16 W. R. Wade, An Introduction to Analysis, third edition, Prentice Hall, 2004.
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.   DOI   ScienceOn
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.   DOI   ScienceOn
19 A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer-Verlag, Berlin, Heidelberg, 1994.
20 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.   DOI