Inverse du Laplacien discret dans le problème de Poisson-Dirichlet à deux dimensions sur un rectangle
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 3, p. 485-552

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.

Ce travail a pour objet l’étude d’une 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 l’opérateur Laplacien par une matrice de Toeplitz à blocs, eux-mêmes de Toeplitz, et nous établissons une formule donnant les blocs de l’inverse 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 l’inverse, de type ergodique, nous passons du discret au continu, en retrouvant l’expression 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.

@article{AFST_2006_6_15_3_485_0,
     author = {Chanzy, Jean},
     title = {Inverse du Laplacien discret dans le probl\`eme de Poisson-Dirichlet \`a deux dimensions sur un rectangle},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {6e s{\'e}rie, 15},
     number = {3},
     year = {2006},
     pages = {485-552},
     doi = {10.5802/afst.1128},
     mrnumber = {2246413},
     zbl = {pre05176317},
     language = {fr},
     url = {http://www.numdam.org/item/AFST_2006_6_15_3_485_0}
}
Chanzy, Jean. Inverse du Laplacien discret dans le problème de Poisson-Dirichlet à deux dimensions sur un rectangle. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 3, pp. 485-552. doi : 10.5802/afst.1128. http://www.numdam.org/item/AFST_2006_6_15_3_485_0/

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