Approximation of the two-dimensional Dirichlet problem by continuous and discrete problems on one-dimensional networks
Confluentes Mathematici, Tome 7 (2015) no. 1, pp. 13-33.

We show that the solution of the two-dimensional Dirichlet problem set in a plane domain is the limit of the solutions of similar problems set on a sequence of one-dimensional networks as their size goes to zero. Roughly speaking this means that a membrane can be seen as the limit of rackets made of strings. For practical applications, we also show that the solutions of the discrete approximated problems (again on the one-dimensional networks) also converge to the solution of the two-dimensional Dirichlet problem.

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DOI : 10.5802/cml.16
Classification : 35R02, 35B40, 65N30
Bourlard-Jospin, Maryse 1 ; Nicaise, Serge 1 ; Venel, Juliette 1

1 Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes, F-59313 Valenciennes Cedex 9, France
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Bourlard-Jospin, Maryse; Nicaise, Serge; Venel, Juliette. Approximation of the two-dimensional Dirichlet problem by continuous and discrete problems on one-dimensional networks. Confluentes Mathematici, Tome 7 (2015) no. 1, pp. 13-33. doi : 10.5802/cml.16. http://www.numdam.org/articles/10.5802/cml.16/

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