AM  Vol.4 No.5 , May 2013
Finite Difference Preconditioners for Legendre Based Spectral Element Methods on Elliptic Boundary Value Problems
ABSTRACT

Finite difference type preconditioners for spectral element discretizations based on Legendre-Gauss-Lobatto points are analyzed. The latter is employed for the approximation of uniformly elliptic partial differential problems. In this work, it is shown that the condition number of the resulting preconditioned system is bounded independently of both of the polynomial degrees used in the spectral element method and the element sizes. Several numerical tests verify the h-p independence of the proposed preconditioning.


Cite this paper
S. Kim, A. St-Cyr and S. Kim, "Finite Difference Preconditioners for Legendre Based Spectral Element Methods on Elliptic Boundary Value Problems," Applied Mathematics, Vol. 4 No. 5, 2013, pp. 838-847. doi: 10.4236/am.2013.45115.
References
[1]   M. O. Deville and E. H. Mund, “Finite Element Preconditioning for Pseudospectral Solutions of Elliptic Problems,” SIAM Journal on Scientific and Statistical Computing, Vol. 11, No. 2, 1990, pp. 311-342. doi:10.1137/0911019

[2]   S. D. Kim and S. Parter, “Preconditioning Legendre spectral Collocation Method for Elliptic Partial Differential Equations,” SIAM Journal on Numerical Analysis, Vol. 33, No. 6, 1996, pp. 2375-2400. doi:10.1137/S0036142994275998

[3]   S. D. Kim and S. Parter, “Preconditioning Chebyshev Spectral Collocation by Finite Difference Operators,” SIAM Journal on Numerical Analysis, Vol. 34, No. 3, 1997, pp. 939-958. doi:10.1137/S0036142995285034

[4]   S. Parter, “Preconditioning Legendre Spectral Collocation Methods for Elliptic Problems I: Finite Difference Operators,” SIAM Journal on Numerical Analysis, Vol. 39, No. 1, 2001, pp. 330-347. doi:10.1137/S0036142999365060

[5]   S. Parter, “Preconditioning Legendre Spectral Collocation Methods for Elliptic Problems II: Finite Element Operators,” SIAM Journal on Numerical Analysis, Vol. 39, No. 1, 2001, pp. 348-362. doi:10.1137/S0036142999365072

[6]   S. Parter and E. Rothman, “Preconditioning Legendre Spectral Collocation Approximations to Elliptic Problems,” SIAM Journal on Numerical Analysis, Vol. 32, No. 2, 1995, pp. 333-385. doi:10.1137/0732015

[7]   A. Quarteroni and E. Zampieri, “Finite Element Preconditioning for Legendre Spectral Collocation Approximations to Elliptic Equations and Systems,” SIAM Journal on Numerical Analysis, Vol. 29, No. 4, 1992, pp. 917-936. doi:10.1137/0729056

[8]   S. A. Orszag, “Spectral Methods for Problems in Complex Geometries,” Journal of Computational Physics, Vol. 37, No. 1, 1980, pp. 70-92. doi:10.1016/0021-9991(80)90005-4

[9]   J. Heys, T. Manteuffel, S. McCormick and L. Olson, “Algebraic Multigrid (AMG) for High-Order Finite Elements,” Journal of Computational Physics, Vol. 204, No. 2, 2005, pp. 520-532. doi:10.1016/j.jcp.2004.10.021

[10]   P. Fischer, “An Overlapping Schwarz Method for Spectral Element Solution of the Incompressible Navier-Stokes equations,” Journal of Computational Physics, Vol. 133, No. 1, 1997, pp. 84-101. doi:10.1006/jcph.1997.5651

[11]   J. W. Lottes and P. F. Fischer, “Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method,” Journal of Scientific Computing, Vol. 24, No. 1, 2005, pp. 45-78. doi:10.1007/s10915-004-4787-3

[12]   S. D. Kim, “Piecewise Bilinear Preconditioning on High Order Finite Element Methods,” Electronic Transactions on Numerical Analysis, Vol. 26, 2007, pp. 228-242.

[13]   S. Kim and S. D. Kim, “Preconditioning on High-Order Element Methods Using Chebyshev-Gauss-Lobatto Nodes,” Applied Numerical Mathematics, Vol. 59, No. 2, 2009, pp. 316-333. doi:10.1016/j.apnum.2008.02.007

[14]   C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, “Spectral Methods: Fundamentals in Single Domains,” Springer, New York, 2006.

[15]   C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, “Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics,” Springer, New York, 2007.

[16]   M. O. Deville, P. F. Fischer and E. H. Mund, “High-Order Methods for Incompressible Fluid Flow,” In: Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2002.

 
 
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