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.
Maryse Bourlard-Jospin 1; Serge Nicaise 1; Juliette Venel 1
@article{CML_2015__7_1_13_0, author = {Maryse Bourlard-Jospin and Serge Nicaise and Juliette Venel}, title = {Approximation of the two-dimensional {Dirichlet} problem by continuous and discrete problems on one-dimensional networks}, journal = {Confluentes Mathematici}, pages = {13--33}, publisher = {Institut Camille Jordan}, volume = {7}, number = {1}, year = {2015}, doi = {10.5802/cml.16}, language = {en}, url = {https://cml.centre-mersenne.org/articles/10.5802/cml.16/} }
TY - JOUR AU - Maryse Bourlard-Jospin AU - Serge Nicaise AU - Juliette Venel TI - Approximation of the two-dimensional Dirichlet problem by continuous and discrete problems on one-dimensional networks JO - Confluentes Mathematici PY - 2015 SP - 13 EP - 33 VL - 7 IS - 1 PB - Institut Camille Jordan UR - https://cml.centre-mersenne.org/articles/10.5802/cml.16/ DO - 10.5802/cml.16 LA - en ID - CML_2015__7_1_13_0 ER -
%0 Journal Article %A Maryse Bourlard-Jospin %A Serge Nicaise %A Juliette Venel %T Approximation of the two-dimensional Dirichlet problem by continuous and discrete problems on one-dimensional networks %J Confluentes Mathematici %D 2015 %P 13-33 %V 7 %N 1 %I Institut Camille Jordan %U https://cml.centre-mersenne.org/articles/10.5802/cml.16/ %R 10.5802/cml.16 %G en %F CML_2015__7_1_13_0
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, Volume 7 (2015) no. 1, pp. 13-33. doi : 10.5802/cml.16. https://cml.centre-mersenne.org/articles/10.5802/cml.16/
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