Une méthode est développée pour la construction des solutions de l’équation dans , soumises aux conditions aux limites , sur . Le problème se réduit à celui de trouver la projection orthogonale de sur le sous-espace des fonctions harmoniques dans . Ce dernier problème est résolu par la décomposition de en somme directe (non orthogonale) de deux sous-espaces fermés pour lesquels des bases orthonormées complètes sont connues explicitement. La projection est exprimée en termes des projections , , de sur . Ceci permet d’établir une méthode d’approximation pour les solutions 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 in , subject to boundary conditions , on . The problem is reduced to that of finding the orthogonal projection of in onto the subspace of square integrable functions harmonic in . This problem is solved by decomposition into the closed direct (not orthogonal) sum of two subspaces for which complete orthogonal bases are known. is expressed in terms of the projections , of onto , 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}, address = {Grenoble}, volume = {23}, number = {3}, year = {1973}, doi = {10.5802/aif.472}, mrnumber = {50 #760}, zbl = {0258.31009}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.472/} }
TY - JOUR AU - Aronszajn, Nachman AU - Brown, R. D. AU - Butcher, R. S. TI - Construction of the solutions of boundary value problems for the biharmonic operator in a rectangle JO - Annales de l'Institut Fourier PY - 1973 SP - 49 EP - 89 VL - 23 IS - 3 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/aif.472/ DO - 10.5802/aif.472 LA - en ID - AIF_1973__23_3_49_0 ER -
%0 Journal Article %A Aronszajn, Nachman %A Brown, R. D. %A Butcher, R. S. %T Construction of the solutions of boundary value problems for the biharmonic operator in a rectangle %J Annales de l'Institut Fourier %D 1973 %P 49-89 %V 23 %N 3 %I Institut Fourier %C Grenoble %U http://www.numdam.org/articles/10.5802/aif.472/ %R 10.5802/aif.472 %G en %F 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, 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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