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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, 7 no. 1 (2015), p. 13-33, doi: 10.5802/cml.16
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Class. Math.: 35R02, 35B40, 65N30

Résumé - Abstract

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.


[1] F. Ali Mehmeti. A characterisation of generalized $c^{\infty }$ notion on nets. Integral Eq. and Operator Theory, 9:753–766, 1986.  MR 866963 |  Zbl 0617.35022
[2] F. Ali Mehmeti. Nonlinear wave in networks, volume 80 of Math. Res. Akademie Verlag, 1994.  MR 1287844 |  Zbl 0831.35096
[3] O. Axelsson. Iterative solution methods. Cambridge University Press, Cambridge, 1994.  MR 1276069 |  Zbl 0845.65011
[4] J. von Below. A characteristic equation associated to an eigenvalue problem on $c^2$-networks. Linear Algebra and Appl., 71:309–325, 1985.  MR 813056 |  Zbl 0617.34010
[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.  MR 520174 |  Zbl 0511.65078
[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.  MR 1071376 |  Zbl 0706.73046
[8] D. Cioranescu and J. Saint Jean Paulin. Homogenization of reticulated structures, volume 136 of Applied Mathematical Sciences. Springer-Verlag, New York, 1999.  MR 1676922 |  Zbl 0929.35002
[9] P. Destuynder and M. Salaun. Mathematical analysis of thin plate models, volume 24 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer-Verlag, Berlin, 1996.  MR 1422248 |  Zbl 0860.73001
[10] FreeFEM++ finite element programming environment. http://www.freefem.org/ff++/.
[11] P. Grisvard. Elliptic problems in nonsmooth domains, volume 24 of Monographs and Studies in Mathematics. Pitman, Boston–London–Melbourne, 1985.  MR 775683 |  Zbl 0695.35060
[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.  MR 2152596 |  Zbl 1089.74025
[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.  MR 1782522 |  Zbl 0958.35091
[14] P. Lascaux and R. Théodor. Analyse numérique matricielle appliquée à l’art de l’ingénieur. Tome 1. Masson, Paris, 1986.  MR 835440 |  Zbl 0601.65016
[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.  MR 587842 |  Zbl 0437.35037
[16] G. Lumer. Espaces ramifiés, et diffusions sur les réseaux topologiques. C. R. Acad. Sci. Paris Sér. A-B, 291(12):A627–A630, 1980.  MR 606449 |  Zbl 0449.35110
[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.  MR 921561 |  Zbl 0644.35076
[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.  MR 1785593 |  Zbl 0989.35100
[20] G. Panasenko. Multi-scale modelling for structures and composites. Springer, Dordrecht, 2005.  MR 2133084 |  Zbl 1078.74002
[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.  MR 1751750 |  Zbl 0957.65001
[23] A. Quarteroni and A. Valli. Numerical approximation of partial differential equations, volume 23 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 1994.  MR 1299729 |  Zbl 0803.65088
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