Construction of the solutions of boundary value problems for the biharmonic operator in a rectangle
Annales de l'Institut Fourier, Volume 23 (1973) no. 3, p. 49-89

A technique is developed for constructing the solution of Δ 2 u=F in R={(x,y):|x|<a,|y|<b}, subject to boundary conditions u=ϕ, u n=ψ on R. The problem is reduced to that of finding the orthogonal projection Pw of w in L 2 (R) onto the subspace H of square integrable functions harmonic in R. This problem is solved by decomposition H into the closed direct (not orthogonal) sum of two subspaces H (1) ,H (2) for which complete orthogonal bases are known. P is expressed in terms of the projections P (1) , P (2) of L 2 (R) onto H (1) , H (2) respectively. The resulting construction yields an approximation technique with both a priori and a posteriori error bounds (the latter very precise). In a short appendix the numerical results are given of the application of the technique in some specific examples and the a posteriori error evaluated.

Une méthode est développée pour la construction des solutions de l’équation Δ 2 u=F dans R={(x,y):|x|<a,|y|<b}, soumises aux conditions aux limites u=ϕ, u n=ψ sur R. Le problème se réduit à celui de trouver la projection orthogonale Pw de wL 2 (R) sur le sous-espace HL 2 (R) des fonctions harmoniques dans R. Ce dernier problème est résolu par la décomposition de H en somme directe (non orthogonale) de deux sous-espaces fermés H (1) ,H (2) pour lesquels des bases orthonormées complètes sont connues explicitement. La projection P est exprimée en termes des projections P (i) , i=1,2, de L 2 (R) sur H (i) . Ceci permet d’établir une méthode d’approximation pour les solutions u du problème original admettant des évaluations a priori et a posteriori (celle-ci très précise) de l’erreur. Dans un appendice des résultats numériques sont donnés concernant l’application de la méthode dans quelques cas concrets en utilisant l’évaluation a posteriori de l’erreur.

@article{AIF_1973__23_3_49_0,
     author = {Aronszajn, Nachman and Brown, R. D. and Butcher, R. S.},
     title = {Construction of the solutions of boundary value problems for the biharmonic operator in a rectangle},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {23},
     number = {3},
     year = {1973},
     pages = {49-89},
     doi = {10.5802/aif.472},
     zbl = {0258.31009},
     mrnumber = {50 \#760},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1973__23_3_49_0}
}
Aronszajn, Nachman; Brown, R. D.; Butcher, R. S. Construction of the solutions of boundary value problems for the biharmonic operator in a rectangle. Annales de l'Institut Fourier, Volume 23 (1973) no. 3, pp. 49-89. doi : 10.5802/aif.472. http://www.numdam.org/item/AIF_1973__23_3_49_0/

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