Spectral methods and a maximum principle

Author:
Claudio Canuto

Journal:
Math. Comp. **51** (1988), 615-629

MSC:
Primary 65N30; Secondary 65N35

DOI:
https://doi.org/10.1090/S0025-5718-1988-0930226-2

MathSciNet review:
930226

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Abstract: Various spectral Chebyshev approximations of a model boundary layer problem for both a Helmholtz and an advection-diffusion operator are considered. It is assumed that simultaneously the boundary layer width tends to zero and the resolution power of the numerical method tends to infinity. The behavior of the spectral solutions in the frequency space and in the physical space is investigated. Error estimates are derived.

**[1]**C. Canuto,*Spectral Methods and a Maximum Principle*, IAN-CNR Report n. 476, Pavia, Italy, 1986.**[2]**C. Canuto & D. Funaro, "The Schwarz algorithm for spectral methods,"*SIAM J. Numer. Anal.*, v. 25, 1988, pp. 24-40. MR**923923 (89b:65286)****[3]**C. Canuto, M. Y. Hussaini, A. Quarteroni & T. Zang,*Spectral Methods in Fluid Dynamics*, Springer Series in Computational Physics, Springer-Verlag, New York, 1987. MR**917480 (89m:76004)****[4]**C. Canuto & A. Quarteroni, "Spectral and pseudo-spectral methods for parabolic problems with nonperiodic boundary conditions,"*Calcolo*, v. 18, 1981, pp. 197-217. MR**647825 (84h:35132)****[5]**C. Canuto & G. Sacchi Landriani, "Analysis of the Kleiser-Schumann method,"*Numer. Math.*, v. 50, 1986, pp. 217-243. MR**866138 (88f:65191)****[6]**Ph. G. Ciarlet,*The Finite Element Method for Elliptic Problems*, North-Holland, Amsterdam, 1978. MR**0520174 (58:25001)****[7]**W. Eckhaus,*Singular Perturbations*, North-Holland, Amsterdam, 1973.**[8]**D. Gottlieb & S. A. Orszag,*Numerical Analysis of Spectral Methods*:*Theory and Applications*, SIAM, Philadelphia, Pa., 1977. MR**0520152 (58:24983)****[9]**H. O Kreiss & J. Oliger, "Stability of the Fourier method,"*SIAM J. Numer. Anal.*, v. 16, 1979, pp. 421-433. MR**530479 (80i:65130)****[10]**J. L. Lions,*Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal*, Lecture Notes in Math., Vol. 323, Springer-Verlag, Berlin, 1973. MR**0600331 (58:29078)****[11]**A. Majda, J. McDonough & S. Osher, "The Fourier method for nonsmooth initial data,"*Math. Comp.*, v. 32, 1978, pp. 1041-1081. MR**501995 (80a:65197)****[12 Y]**Maday & A. Quarteroni, "Legendre and Chebyshev spectral approximations of Burgers' equation,"*Numer. Math.*, v. 37, 1981, pp. 321-332. MR**627106 (83c:65246)****[13]**S. A. Orszag & M. Y Israeli, "Numerical simulation of viscous incompressible flows,"*Ann. Rev. Fluid Mech.*, v. 6, 1974, pp. 281-318.**[14]**R. G. Voigt, D. Gottlieb & M. Y. Hussaini (Eds.),*Spectral Methods for Partial Differential Equations*, SIAM, Philadelphia, Pa., 1984. MR**758260 (85g:76003)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1988-0930226-2

Article copyright:
© Copyright 1988
American Mathematical Society