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.

Reçu le : 2014-10-19
Révisé le : 2015-02-18
Accepté le : 2015-02-18
Publié le : 2016-02-03
DOI : https://doi.org/10.5802/cml.16
Classification : 35R02,  35B40,  65N30
@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},
     publisher = {Institut Camille Jordan},
     volume = {7},
     number = {1},
     year = {2015},
     pages = {13-33},
     doi = {10.5802/cml.16},
     language = {en},
     url = {cml.centre-mersenne.org/item/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, Tome 7 (2015) no. 1, pp. 13-33. doi : 10.5802/cml.16. https://cml.centre-mersenne.org/item/CML_2015__7_1_13_0/

[1] F. Ali Mehmeti. A characterisation of generalized c notion on nets. Integral Eq. and Operator Theory, 9:753–766, 1986.

[2] F. Ali Mehmeti. Nonlinear wave in networks, volume 80 of Math. Res. Akademie Verlag, 1994.

[3] O. Axelsson. Iterative solution methods. Cambridge University Press, Cambridge, 1994.

[4] J. von Below. A characteristic equation associated to an eigenvalue problem on c 2 -networks. Linear Algebra and Appl., 71:309–325, 1985.

[5] J. von Below. Parabolic network equations. Technical report, Eberhard-Karls-Universität Tübingen, 1993. Habilitationsschrift.

[6] P. G. Ciarlet. The finite element method for elliptic problems. North-Holland, Amsterdam, 1978.

[7] P. G. Ciarlet. Plates and junctions in elastic multi-structures, volume 14 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics]. Masson, Paris, 1990. An asymptotic analysis.

[8] D. Cioranescu and J. Saint Jean Paulin. Homogenization of reticulated structures, volume 136 of Applied Mathematical Sciences. Springer-Verlag, New York, 1999.

[9] P. Destuynder and M. Salaun. Mathematical analysis of thin plate models, volume 24 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer-Verlag, Berlin, 1996.

[10] FreeFEM++ finite element programming environment. .

[11] P. Grisvard. Elliptic problems in nonsmooth domains, volume 24 of Monographs and Studies in Mathematics. Pitman, Boston–London–Melbourne, 1985.

[12] A. V. Komarov and O. M. Penkin. On the spectrum of a nonperiodic woven membrane. Sovrem. Mat. Prilozh., (16, Differ. Uravn. Chast. Proizvod.):3–21, 2004.

[13] A. V. Komarov, O. M. Penkin, and Y. V. Pokornyĭ. On the spectrum of a uniform network of strings. Izv. Vyssh. Uchebn. Zaved. Mat., (4):23–27, 2000.

[14] P. Lascaux and R. Théodor. Analyse numérique matricielle appliquée à l’art de l’ingénieur. Tome 1. Masson, Paris, 1986.

[15] G. Lumer. Connecting of local operators and evolution equations on networks. In Potential theory, Copenhagen 1979 (Proc. Colloq., Copenhagen, 1979), volume 787 of Lecture Notes in Math., pages 219–234. Springer, Berlin, 1980.

[16] G. Lumer. Espaces ramifiés, et diffusions sur les réseaux topologiques. C. R. Acad. Sci. Paris Sér. A-B, 291(12):A627–A630, 1980.

[17] S. Nicaise. Diffusion sur les espaces ramifiés. PhD thesis, U. Mons (Belgium), 1986.

[18] S. Nicaise. Spectre des réseaux topologiques finis. Bull. Sc. Math., 2ème série, 111:401–413, 1987.

[19] S. Nicaise and O. Penkin. Relationship between the lower frequency spectrum of plates and networks of beams. Math. Methods Appl. Sci., 23(16):1389–1399, 2000.

[20] G. Panasenko. Multi-scale modelling for structures and composites. Springer, Dordrecht, 2005.

[21] O. Penkin. Some qualitative properties of the boundary values problems on graphs. PhD thesis, U. Voronezh (Russia), 1988.

[22] A. Quarteroni, R. Sacco, and F. Saleri. Numerical mathematics, volume 37 of Texts in Applied Mathematics. Springer-Verlag, New York, 2000.

[23] A. Quarteroni and A. Valli. Numerical approximation of partial differential equations, volume 23 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 1994.