Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T19:21:13.162Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite-difference preconditioners for superconsistent pseudospectral approximations

Published online by Cambridge University Press:  15 December 2007

Lorella Fatone ,
Daniele Funaro  and
Valentina Scannavini
Lorella Fatone
Affiliation:
Dipartimento di Matematica Pura ed Applicata, via Campi 213/b, Università di Modena e Reggio Emilia, Modena 41110, Italy. lorella.fatone@unimo.it; daniele.funaro@unimo.it; s.valentina@tin.it
Daniele Funaro
Affiliation:
Dipartimento di Matematica Pura ed Applicata, via Campi 213/b, Università di Modena e Reggio Emilia, Modena 41110, Italy. lorella.fatone@unimo.it; daniele.funaro@unimo.it; s.valentina@tin.it
Valentina Scannavini
Affiliation:
Dipartimento di Matematica Pura ed Applicata, via Campi 213/b, Università di Modena e Reggio Emilia, Modena 41110, Italy. lorella.fatone@unimo.it; daniele.funaro@unimo.it; s.valentina@tin.it
Get access

Abstract

The superconsistent collocation method, which is based on acollocation grid different from the one used to represent thesolution, has proven to be very accurate in the resolution ofvarious functional equations. Excellent results can be alsoobtained for what concerns preconditioning. Some analysis andnumerous experiments, regarding the use of finite-differencespreconditioners, for matrices arising from pseudospectralapproximations of advection-diffusion boundary value problems, arepresented and discussed, both in the case of Legendre andChebyshev representation nodes.

Keywords

Spectral collocation methodpreconditioningsuperconsistencyLebesgue constant.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 6 , November 2007 , pp. 1021 - 1039
Copyright
© EDP Sciences, SMAI, 2007

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

Canuto, C. and Pietra, P., Boundary interface conditions within a finite element preconditioner for spectral methods. J. Comput. Phys. 91 (1990) 310–343. CrossRef
Canuto, C. and Quarteroni, A., Spectral and pseudo-spectral methods for parabolic problems with nonperiodic boundary conditions. Calcolo 18 (1981) 197–218. CrossRef
C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods in Fluid Dynamics. Springer, New York (1988).
L. Fatone, D. Funaro and G.J. Yoon, A convergence analysis for the superconsistent Chebyshev method. Appl. Num. Math. (2007) (to appear).
D. Funaro, Polynomial Approximation of Differential Equations, Lecture Notes in Physics 8. Springer, Heidelberg (1992).
Funaro, D., Some remarks about the collocation method on a modified Legendre grid. J. Comput. Appl. Math. 33 (1997) 95–103. CrossRef
D. Funaro, Spectral Elements for Transport-Dominated Equations, Lecture Notes in Computational Science and Engineering 1. Springer (1997).
Funaro, D., A superconsistent Chebyshev collocation method for second-order differential operators. Numer. Algorithms 28 (2001) 151–157. CrossRef
Funaro, D., Superconsistent discretizations. J. Scientific Computing 17 (2002) 67–80. CrossRef
D. Gottlieb, M.Y. Hussaini and S.A. Orszag, Theory and application of spectral methods, in Spectral Methods for Partial Differential Equations, R.G. Voigt, D. Gottlieb and M.Y. Hussaini Eds., SIAM, Philadelphia (1984).
Haldenwang, P., Labrosse, G., Abboudi, S. and Deville, M., Chebyshev 3-D spectral and 2-D pseudospectral solvers for the Helhmoltz equation. J. Comput. Phys. 55 (1981) 115–128. CrossRef
Kilgore, T., A characterization of the Lagrange interpolation projections with minimal Tchebycheff norm. J. Approximation Theory 24 (1978) 273–288. CrossRef
Kim, D.H., Kwon, K.H., Marcellán, F. and Park, S.B., Fourier, On series of a discrete Jacobi-Sobolev inner product. J. Approximation Theory 117 (2002) 1–22.
Kim, S.D. and Parter, S.V., Preconditioning Chebyshev spectral collocation method for elliptic partial differential equations. SIAM J. Numer. Anal. 33 (1996) 2375–2400. CrossRef
Kim, S.D. and Parter, S.V., Preconditioning Chebyshev spectral collocation by finite-difference operators. SIAM J. Numer. Anal. 34 (1997) 939–958. CrossRef
Marcellán, F., Osilenker, B.P. and Rocha, I.A., Sobolev-type orthogonal polynomials and their zeros. Rendiconti di Matematica 17 (1997) 423–444.
Mund, E.H., A short survey on preconditioning techniques in spectral calculations. Appl. Num. Math. 33 (2000) 61–70. CrossRef
Orszag, S.A., Spectral methods for problems in complex geometries. J. Comput. Phys. 37 (1980) 70–92. CrossRef
G. Szegö, Orthogonal Polynomials. American Mathematical Society, New York (1939).
L.N. Trefethen and M. Embree, Spectra and Pseudospectra: the behavior of nonnormal matrices and operators. Princeton University Press (2005).