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Some spectral approximations of two-dimensional fourth-order problems


Authors: Christine Bernardi, Giuseppe Coppoletta and Yvon Maday
Journal: Math. Comp. 59 (1992), 63-76
MSC: Primary 65N35; Secondary 65D30, 76D05, 76M25
MathSciNet review: 1134714
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Abstract: This paper deals with the approximation of the biharmonic equation in a square domain with Dirichlet boundary conditions. Two types of discrete problems are presented, the numerical analysis is performed, and estimates for the error between the exact and approximate solutions are given.


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  • [1] C. Bernardi, G. Coppoletta, and Y. Maday, Some spectral approximations of multidimensional fourth-order problems, Internal Report 90021, Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie, Paris, 1990.
  • [2] Christine Bernardi and Yvon Maday, Some spectral approximations of one-dimensional fourth-order problems, Progress in approximation theory, Academic Press, Boston, MA, 1991, pp. 43–116. MR 1114769
  • [3] -, Spectral methods, Handbook of Numerical Analysis (P. G. Ciarlet and J.-L. Lions, eds.), North-Holland (to appear).
  • [4] C. Bernardi and G. Raugel, Méthodes d’éléments finis mixtes pour les équations de Stokes et de Navier-Stokes dans un polygone non convexe, Calcolo 18 (1981), no. 3, 255–291 (French, with English summary). MR 647827, 10.1007/BF02576359
  • [5] Claudio Canuto, M. Yousuff Hussaini, Alfio Quarteroni, and Thomas A. Zang, Spectral methods in fluid dynamics, Springer Series in Computational Physics, Springer-Verlag, New York, 1988. MR 917480
  • [6] C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp. 38 (1982), no. 157, 67–86. MR 637287, 10.1090/S0025-5718-1982-0637287-3
  • [7] Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. MR 760629
  • [8] Daniele Funaro and Wilhelm Heinrichs, Some results about the pseudospectral approximation of one-dimensional fourth-order problems, Numer. Math. 58 (1990), no. 4, 399–418. MR 1077586, 10.1007/BF01385633
  • [9] Luigi Gatteschi, Su una formula di quadratura “quasi gaussiana”. Tabulazione delle ascisse d’integrazione e delle relative costanti di Christoffel, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 98 (1963/1964), 641–661 (Italian). MR 0166920
  • [10] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
  • [11] E. B. Karpilovskaya, A method of collocation for integro-differential equations with biharmonic principal part, U.S.S.R. Comput. Math. and Math. Phys. 10 (1970), no. 6, 240-246.
  • [12] O. Koutchmy (in preparation).
  • [13] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243
  • [14] Yvon Maday, Résultats d’approximation optimaux pour les opérateurs d’interpolation polynomiale, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 9, 705–710 (French, with English summary). MR 1105630
  • [15] 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
  • [16] S. A. Orszag, Accurate solution of the Orr-Sommerfeld stability equation, J. Fluid Mech. 50 (1971), 689-703.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1992-1134714-9
Article copyright: © Copyright 1992 American Mathematical Society