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

Une méthode est développée pour la construction des solutions de l’équation ${\Delta }^{2}u=F$ dans $R=\left\{\left(x,y\right):|x|, soumises aux conditions aux limites $u=\varphi$, $\frac{\partial u}{\partial n}=\psi$ sur $\partial R$. Le problème se réduit à celui de trouver la projection orthogonale $Pw$ de $w\in {L}^{2}\left(R\right)$ sur le sous-espace $\mathbf{H}\subset {L}^{2}\left(R\right)$ des fonctions harmoniques dans $R$. Ce dernier problème est résolu par la décomposition de $\mathbf{H}$ en somme directe (non orthogonale) de deux sous-espaces fermés ${\mathbf{H}}^{\left(1\right)},{\mathbf{H}}^{\left(2\right)}$ pour lesquels des bases orthonormées complètes sont connues explicitement. La projection $P$ est exprimée en termes des projections ${P}^{\left(i\right)}$, $i=1,2$, de ${L}^{2}\left(R\right)$ sur ${\mathbf{H}}^{\left(i\right)}$. 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.

A technique is developed for constructing the solution of ${\Delta }^{2}u=F$ in $R=\left\{\left(x,y\right):|x|, subject to boundary conditions $u=\varphi$, $\frac{\partial u}{\partial n}=\psi$ on $\partial R$. The problem is reduced to that of finding the orthogonal projection $Pw$ of $w$ in ${L}^{2}\left(R\right)$ onto the subspace $\mathbf{H}$ of square integrable functions harmonic in $\mathbf{R}$. This problem is solved by decomposition $\mathbf{H}$ into the closed direct (not orthogonal) sum of two subspaces ${\mathbf{H}}^{\left(1\right)},{\mathbf{H}}^{\left(2\right)}$ for which complete orthogonal bases are known. $P$ is expressed in terms of the projections ${P}^{\left(1\right)}$, ${P}^{\left(2\right)}$ of ${L}^{2}\left(R\right)$ onto ${\mathbf{H}}^{\left(1\right)}$, ${\mathbf{H}}^{\left(2\right)}$ 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.

@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},
pages = {49--89},
publisher = {Institut Fourier},
volume = {23},
number = {3},
year = {1973},
doi = {10.5802/aif.472},
zbl = {0258.31009},
mrnumber = {50 #760},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.472/}
}
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, Tome 23 (1973) no. 3, pp. 49-89. doi : 10.5802/aif.472. http://www.numdam.org/articles/10.5802/aif.472/

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