Nouveau noyau de Green associé au problème de Poisson-Dirichlet sur un rectangle

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).