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Some spectral approximations of two-dimensional fourth-order problems. (English) Zbl 0754.65088

The biharmonic equation \(\Delta^ 2u=f\) in a square domain \(\Omega\) with Dirichlet boundary conditions and continuous function \(f\) is considered as a model problem. Two discretizations of spectral type are considered with the aim of finding an approximate solution in a finite-dimensional space of polynomials.
The first method is of collocation type based on the Gauss-Lobatto nodes and weights of the corresponding quadrature rule with end points of multiplicity 2. It is shown that the discrete problem is equivalent to a variational formulation, from which the uniqueness of the solution can be derived. Moreover, an error estimate is derived for this natural and simple approach that is, however, not optimal.
Therefore a second discretization is proposed, which is no longer of collocation type, but for which optimal error bounds are proved for two special cases of nodes and weights.

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
35J40 Boundary value problems for higher-order elliptic equations
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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. · Zbl 0701.41009
[3] -, Spectral methods, Handbook of Numerical Analysis , 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). · Zbl 0475.76035 · doi: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. · Zbl 0658.76001
[6] C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp. 38 (1982), no. 157, 67 – 86. · Zbl 0567.41008
[7] Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. · Zbl 0537.65020
[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. · Zbl 0714.65074 · doi: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). · Zbl 0135.38403
[10] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. · Zbl 0695.35060
[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). · Zbl 0212.43801
[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). · Zbl 0724.65004
[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). · Zbl 0607.76024
[16] S. A. Orszag, Accurate solution of the Orr-Sommerfeld stability equation, J. Fluid Mech. 50 (1971), 689-703. · Zbl 0237.76027
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