Finite-difference preconditioners for superconsistent pseudospectral approximations
ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 6, pp. 1021-1039.

The superconsistent collocation method, which is based on a collocation grid different from the one used to represent the solution, has proven to be very accurate in the resolution of various functional equations. Excellent results can be also obtained for what concerns preconditioning. Some analysis and numerous experiments, regarding the use of finite-differences preconditioners, for matrices arising from pseudospectral approximations of advection-diffusion boundary value problems, are presented and discussed, both in the case of Legendre and Chebyshev representation nodes.

DOI : 10.1051/m2an:2007052
Classification : 65N35, 65F15, 41A10
Mots-clés : spectral collocation method, preconditioning, superconsistency, Lebesgue constant 
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     title = {Finite-difference preconditioners for superconsistent pseudospectral approximations},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {1021--1039},
     publisher = {EDP-Sciences},
     volume = {41},
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     year = {2007},
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     zbl = {1133.65103},
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Fatone, Lorella; Funaro, Daniele; Scannavini, Valentina. Finite-difference preconditioners for superconsistent pseudospectral approximations. ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 6, pp. 1021-1039. doi : 10.1051/m2an:2007052. http://www.numdam.org/articles/10.1051/m2an:2007052/

[1] C. Canuto and P. Pietra, Boundary interface conditions within a finite element preconditioner for spectral methods. J. Comput. Phys. 91 (1990) 310-343. | Zbl

[2] C. Canuto and A. Quarteroni, Spectral and pseudo-spectral methods for parabolic problems with nonperiodic boundary conditions. Calcolo 18 (1981) 197-218. | Zbl

[3] C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods in Fluid Dynamics. Springer, New York (1988). | MR | Zbl

[4] L. Fatone, D. Funaro and G.J. Yoon, A convergence analysis for the superconsistent Chebyshev method. Appl. Num. Math. (2007) (to appear). | MR

[5] D. Funaro, Polynomial Approximation of Differential Equations, Lecture Notes in Physics 8. Springer, Heidelberg (1992). | MR | Zbl

[6] D. Funaro, Some remarks about the collocation method on a modified Legendre grid. J. Comput. Appl. Math. 33 (1997) 95-103. | Zbl

[7] D. Funaro, Spectral Elements for Transport-Dominated Equations, Lecture Notes in Computational Science and Engineering 1. Springer (1997). | MR | Zbl

[8] D. Funaro, A superconsistent Chebyshev collocation method for second-order differential operators. Numer. Algorithms 28 (2001) 151-157. | Zbl

[9] D. Funaro, Superconsistent discretizations. J. Scientific Computing 17 (2002) 67-80. | Zbl

[10] 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). | Zbl

[11] P. Haldenwang, G. Labrosse, S. Abboudi and M. Deville, Chebyshev 3-D spectral and 2-D pseudospectral solvers for the Helhmoltz equation. J. Comput. Phys. 55 (1981) 115-128. | Zbl

[12] T. Kilgore, A characterization of the Lagrange interpolation projections with minimal Tchebycheff norm. J. Approximation Theory 24 (1978) 273-288. | Zbl

[13] D.H. Kim, K.H. Kwon, F. Marcellán and S.B. Park, On Fourier series of a discrete Jacobi-Sobolev inner product. J. Approximation Theory 117 (2002) 1-22. | Zbl

[14] S.D. Kim and S.V. Parter, Preconditioning Chebyshev spectral collocation method for elliptic partial differential equations. SIAM J. Numer. Anal. 33 (1996) 2375-2400. | Zbl

[15] S.D. Kim and S.V. Parter, Preconditioning Chebyshev spectral collocation by finite-difference operators. SIAM J. Numer. Anal. 34 (1997) 939-958. | Zbl

[16] F. Marcellán, B.P. Osilenker and I.A. Rocha, Sobolev-type orthogonal polynomials and their zeros. Rendiconti di Matematica 17 (1997) 423-444. | Zbl

[17] E.H. Mund, A short survey on preconditioning techniques in spectral calculations. Appl. Num. Math. 33 (2000) 61-70. | Zbl

[18] S.A. Orszag, Spectral methods for problems in complex geometries. J. Comput. Phys. 37 (1980) 70-92. | Zbl

[19] G. Szegö, Orthogonal Polynomials. American Mathematical Society, New York (1939). | JFM | Zbl

[20] L.N. Trefethen and M. Embree, Spectra and Pseudospectra: the behavior of nonnormal matrices and operators. Princeton University Press (2005). | MR | Zbl

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