Analysis of spectral projectors in one-dimensional domains
HTML articles powered by AMS MathViewer
- by Y. Maday PDF
- Math. Comp. 55 (1990), 537-562 Request permission
Abstract:
In this paper we analyze a class of projection operators with values in a subspace of polynomials. These projection operators are related to the Hilbert spaces involved in the numerical analysis of spectral methods. They are, in the first part of the paper, the standard Sobolev spaces and, in the second part, some weighted Sobolev spaces, the weight of which is related to the orthogonality relation satisfied by the Chebyshev polynomials. These results are used to study the approximation of a model fourth-order problem.References
- Christine Bernardi and Yvon Maday, Properties of some weighted Sobolev spaces and application to spectral approximations, SIAM J. Numer. Anal. 26 (1989), no. 4, 769–829 (English, with French summary). MR 1005511, DOI 10.1137/0726045 —, Some spectral approximation of one-dimensional fourth-order problems, J. Approx. Theory (to appear).
- Christine Bernardi, Yvon Maday, and Brigitte Métivet, Spectral approximation of the periodic-nonperiodic Navier-Stokes equations, Numer. Math. 51 (1987), no. 6, 655–700. MR 914344, DOI 10.1007/BF01400175
- C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp. 38 (1982), no. 157, 67–86. MR 637287, DOI 10.1090/S0025-5718-1982-0637287-3
- Claudio Canuto and Alfio Quarteroni, Variational methods in the theoretical analysis of spectral approximations, Spectral methods for partial differential equations (Hampton, Va., 1982) SIAM, Philadelphia, PA, 1984, pp. 55–78. MR 758262
- P. Grisvard, Espaces intermédiaires entre espaces de Sobolev avec poids, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 255–296 (French). MR 160104 —, Private communication. J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. I, Dunod, 1968.
- J.-L. Lions and J. Peetre, Sur une classe d’espaces d’interpolation, Inst. Hautes Études Sci. Publ. Math. 19 (1964), 5–68 (French). MR 165343 Y. Maday and B. Métivet, Estimation d’erreur pour l’approximation des équations de Stokes par une méthode spectrale, Rech. Aérospat. 5 (1983), 237-244.
- Y. Maday and B. Métivet, Chebyshev spectral approximation of Navier-Stokes equations in a two-dimensional domain, RAIRO Modél. Math. Anal. Numér. 21 (1987), no. 1, 93–123 (English, with French summary). MR 882688, DOI 10.1051/m2an/1987210100931
- Yvon Maday and Alfio Quarteroni, Legendre and Chebyshev spectral approximations of Burgers’ equation, Numer. Math. 37 (1981), no. 3, 321–332. MR 627106, DOI 10.1007/BF01400311 F. Montigny-Rannou, Etude d’un problème de stabilité par résolution de l’équation de Orr-Sommerfeld. Application à la détermination d’un champs de vitesse initial turbulent, ONERA-RT 4/3419 AY, 1980. J. Necas, Les méthodes directes en théorie des équations elliptiques, Édition de l’Académie Tchécoslovaque des Sciences, Prague, 1967. S. A. Orszag, Accurate solution of the Orr-Sommerfeld stability equation, J. Fluid Mech. 50 (1971), 689-703.
- Alfio Quarteroni, Blending Fourier and Chebyshev interpolation, J. Approx. Theory 51 (1987), no. 2, 115–126. MR 909803, DOI 10.1016/0021-9045(87)90026-8 F. Riesz and B. Sz.-Nagy, Leçons d’analyse fonctionelle, Akad. Kiado, Budapest, 1952.
- Giovanni Sacchi Landriani and Hervé Vandeven, Polynomial approximation of divergence-free functions, Math. Comp. 52 (1989), no. 185, 103–130. MR 971405, DOI 10.1090/S0025-5718-1989-0971405-9
- Eitan Tadmor, The exponential accuracy of Fourier and Chebyshev differencing methods, SIAM J. Numer. Anal. 23 (1986), no. 1, 1–10. MR 821902, DOI 10.1137/0723001
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Math. Comp. 55 (1990), 537-562
- MSC: Primary 41A65; Secondary 65J05, 65N30
- DOI: https://doi.org/10.1090/S0025-5718-1990-1035939-1
- MathSciNet review: 1035939