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Comptes Rendus Mathématique
Volume 341, n° 1
pages 11-14 (juillet 2005)
Doi : 10.1016/j.crma.2005.05.017
Received : 22 February 2005 ;  accepted : 19 May 2005
Nouveau noyau de Green associé au problème de Poisson-Dirichlet sur un rectangle
New Green kernel associated to the Poisson-Dirichlet problem on a rectangle
 

Jean Chanzy
Laboratoire de mathématiques, bâtiment 425, université de Paris-sud, 91405 Orsay cedex, France 

Résumé

Cette Note a pour objet lʼétude dʼune méthode de « discrétisation » du laplacien dans le problème de Poisson à deux dimensions sur un rectangle, avec des conditions aux limites de Dirichlet. Nous approchons lʼopérateur laplacien par une matrice de Toeplitz à blocs, eux-mêmes de Toeplitz, et nous établissons une formule donnant les blocs de lʼinverse de cette matrice. Nous donnons ensuite un développement asymptotique de la trace de la matrice inverse, et du déterminant de la matrice de Toeplitz. Enfin, par un passage à la limite dans lʼinverse, de type ergodique, nous passons du discret au continu, en retrouvant lʼexpression connue du noyau de Green du problème de Poisson, sous forme de série, et en en donnant une nouvelle expression asymptotique plus intéressante, car elle converge plus rapidement. Pour citer cet article : J. Chanzy, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

The full text of this article is available in PDF format.
Abstract

This work is focused on the study of a discretization' method for the Laplacian operator, in the two-dimensional Poisson problem on a rectangle, with Dirichlet boundary conditions. The Laplacian operator is approximated by a block Toeplitz matrix, the blocks of which are Toeplitz matrices again, and a formula of the inverse matrix blocks is given. Then an asymptotic development of the inverse matrix trace and the Toeplitz matrix determinant are obtained. Finally, the continuum expression of the Laplacian operator is found by calculating the ergodic limit of the inverse matrix. A new asymptotic formula for the well known Green function for the Poisson problem that we obtain converges more rapidly than the usual one. To cite this article: J. Chanzy, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

The full text of this article is available in PDF format.


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