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On error estimates for Galerkin spectral discretizations of parabolic problems with nonsmooth initial data

Authors: Javier de Frutos and Rafael Muñoz-Sola
Journal: Math. Comp. 70 (2001), 525-531
MSC (2000): Primary 65M70, 65M15
Published electronically: March 1, 2000
MathSciNet review: 1680871
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We analyze the Legendre and Chebyshev spectral Galerkin semidiscretizations of a one dimensional homogeneous parabolic problem with nonconstant coefficients. We present error estimates for both smooth and nonsmooth data. In the Chebyshev case a limit in the order of approximation is established. On the contrary, in the Legendre case we find an arbitrary high order of convegence.

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  • 1. J. Bergh and J. Löfström, Interpolation spaces: an introduction, Springer-Verlag, Berlin Heidelberg, 1976. MR 58:2349
  • 2. C. Bernardi and Y. Maday, Properties of some weighted Sobolev spaces and applications to spectral approximations, SIAM J. Numer. Anal., 26, 1989, 769-829. MR 91c:46046
  • 3. C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral methods in Fluid Dynamics, Springer-Verlag, New York, 1988. MR 89m:76004
  • 4. C. Canuto and A. Quarteroni, Spectral and pseudospectral methods for parabolic problems with non periodic boundary conditions, Calcolo, 18, 1981, 197-217. MR 84h:35132
  • 5. R. Dautray and J. L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Vol. 3 , Masson, Paris, 1985. MR 88i:00003b
  • 6. J. de Frutos and R. Muñoz Sola, Chebyshev pseudospectral collocation for parabolic problems with nonconstant coefficients, Proceedings of the third international conference on spectral and high order methods, Houston (Texas), 1996, 101-107.
  • 7. J. de Frutos and R. Muñoz Sola, Error estimates for Galerkin spectral discretizations of parabolic problems with nonsmooth data, Applied Mathematics and Computation Reports 1998/7, Universidad de Valladolid (Spain), pp. 749-754. CMP 98:15
  • 8. G. Fernández Manín. Algunas contribuciones al estudio del error en los métodos espectrales: optimalidad de los métodos de Jacobi y estudio del método de ``patching". PhD Thesis. Santiago de Compostela, 1995.
  • 9. G. Fernández Manín and R. Muñoz Sola, Polynomial approximation of some singular solutions in weighted Sobolev spaces, Proceedings of the third international conference on spectral and high order methods, Houston (Texas), (1996), 93-99. MR 97g:00019
  • 10. H. Fujita and T. Suzuki, Evolution problems, in Handbook of numerical analysis, Vol. II, P. G. Ciarlet and J. L. Lions eds., North Holland, Amsterdam, 1991, pp. 789-928. MR 92f:65001
  • 11. M. Luskin and R. Rannacher, On the smoothing property of the Galerkin method for parabolic equations, SIAM J. Numer. Anal., 19, 1982, 93-113. MR 83c:65245
  • 12. A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983. MR 85g:47061
  • 13. V. Thomée, Galerkin finite element methods for parabolic problems, Lecture Notes in Math., vol. 1054, Springer-Verlag, Berlin Heidelberg, 1984. MR 86k:65006

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Additional Information

Javier de Frutos
Affiliation: Departamento de Matemática Aplicada y Computación, Universidad de Valladolid, Valladolid, Spain

Rafael Muñoz-Sola
Affiliation: Departamento de Matemática Aplicada, Universidad de Santiago de Compostela, Santiago de Compostela, Spain

Keywords: Spectral Galerkin method, parabolic equation, nonsmooth initial data
Received by editor(s): January 4, 1999
Received by editor(s) in revised form: April 6, 1999
Published electronically: March 1, 2000
Additional Notes: J. de Frutos was partially supported by project DGICYT PB95-705 and project JCyL VA52/96. R. Muñoz-Sola was partially supported by project DGICYT PB96-0952.
Article copyright: © Copyright 2000 American Mathematical Society

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