Approximation of the two-dimensional Dirichlet problem by continuous and discrete problems on one-dimensional networks

Maryse Bourlard-Jospin (1) , Serge Nicaise (1) , Juliette Venel (1)

1 LAMAV - Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015

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.

Domaines: Analyse numérique [math.NA]

Maryse Bourlard-Jospin, Serge Nicaise, Juliette Venel. Approximation of the two-dimensional Dirichlet problem by continuous and discrete problems on one-dimensional networks. Confluentes Mathematici, 2015, 7 (1), pp.13-33. ⟨10.5802/cml.16⟩. ⟨hal-03135676⟩