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Inverse of the discrete Laplacian in the two-dimensional Poisson-Dirichlet problem on a rectangle. (Inverse du Laplacien discret dans le problème de Poisson-Dirichlet à deux dimensions sur un rectangle.) (French) Zbl 1133.47054

The paper addresses the Poisson problem on a rectangle with Dirichlet boundary conditions. The author performs a discretization of the problem by means of a representation of the Laplace operator by a block Toeplitz matrix (where the blocks are also Toeplitz matrices). This allows the reduction of the first problem to a linear problem characterized by a corresponding Toeplitz operator. Within this new framework, the author uses some of his previous results to obtain the inverse matrix of the above mentioned discretized Laplacian as well as an asymptotic representation of the inverse matrix trace and the Toeplitz matrix determinant. Then, the passage from the discrete to the continuous case is done through the ergodic limit of the inverse matrix. By using a Spitzer-Stone type theorem, the known classic representation of the Green function of the Poisson problem on a rectangle is derived. Moreover, a new series representation for this Green function is also found and it is exhibited that this new series converges more rapidly than the classic one.
The results of this 68 pages paper are presented with very detailed proofs, and this work is basically the content of one chapter of the author’s Ph. D. Thesis.

MSC:

47N20 Applications of operator theory to differential and integral equations
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
35C10 Series solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
47G10 Integral operators
65F05 Direct numerical methods for linear systems and matrix inversion
65N06 Finite difference methods for boundary value problems involving PDEs
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References:

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