A multiscale correction method for local singular perturbations of the boundary
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 1, p. 111-127

In this work, we consider singular perturbations of the boundary of a smooth domain. We describe the asymptotic behavior of the solution u ε of a second order elliptic equation posed in the perturbed domain with respect to the size parameter ε of the deformation. We are also interested in the variations of the energy functional. We propose a numerical method for the approximation of u ε based on a multiscale superposition of the unperturbed solution u 0 and a profile defined in a model domain. We conclude with numerical results.

DOI : https://doi.org/10.1051/m2an:2007012
Classification:  35B25,  35B40,  35J25,  49Q10,  65N30
Keywords: multiscale asymptotic analysis, shape optimization, patch of elements
@article{M2AN_2007__41_1_111_0,
     author = {Dambrine, Marc and Vial, Gr\'egory},
     title = {A multiscale correction method for local singular perturbations of the boundary},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {41},
     number = {1},
     year = {2007},
     pages = {111-127},
     doi = {10.1051/m2an:2007012},
     zbl = {1129.65084},
     mrnumber = {2323693},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2007__41_1_111_0}
}
Dambrine, Marc; Vial, Grégory. A multiscale correction method for local singular perturbations of the boundary. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 1, pp. 111-127. doi : 10.1051/m2an:2007012. http://www.numdam.org/item/M2AN_2007__41_1_111_0/

[1] G. Allaire, Shape optimization by the homogenization method, Applied Mathematical Sciences 146. Springer-Verlag, New York (2002). | MR 1859696 | Zbl 0990.35001

[2] D. Brancherie and A. Ibrahimbegović, Modélisation ‘macro' de phénomènes localisés à l'échelle ‘micro' : formulation et implantation numérique. Revue européenne des éléments finis, numéro spécial Giens 2003 13 (2004) 461-473. | Zbl pre05147337

[3] D. Brancherie, M. Dambrine, G. Vial and P. Villon, Ultimate load computation, effect of surfacic defect and adaptative techniques, in 7th World Congress in Computational Mechanics, Los Angeles (2006).

[4] G. Caloz, M. Costabel, M. Dauge and G. Vial, Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer. Asymptotic Anal. 50 (2006) 121-173. | Zbl 1136.35021

[5] M. Dambrine and G. Vial, On the influence of a boundary perforation on the dirichlet energy. Control Cybern. 34 (2005) 117-136.

[6] B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves. Math. Comp. 31 (1977) 629-651. | Zbl 0367.65051

[7] D. Givoli, Nonreflecting boundary conditions. J. Comput. Phys. 94 (1991) 1-29. | Zbl 0731.65109

[8] A.M. Il'Lin, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems. Translations of Mathematical Monographs 102, Amer. Math. Soc., Providence, R.I. (1992). | Zbl 0754.34002

[9] V.A. Kondrat'Ev, Boundary value problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc. 16 (1967) 227-313. | Zbl 0194.13405

[10] D. Leguillon and E. Sanchez-Palencia, Computation of singular solutions in elliptic problems and elasticity. Masson, Paris (1987). | MR 995254 | Zbl 0647.73010

[11] M. Lenoir and A. Tounsi, The localized finite element method and its application to the two-dimensional sea-keeping problem. SIAM J. Numer. Anal. 25 (1988) 729-752. | Zbl 0656.76008

[12] T. Lewiński and J. Sokołowski, Topological derivative for nucleation of non-circular voids. The Neumann problem, in Differential geometric methods in the control of partial differential equations (Boulder, CO, 1999), Contemp. Math. 268, Amer. Math. Soc., Providence, RI (2000) 341-361. | Zbl 1050.49028

[13] M. Masmoudi, The Topological Asymptotic, in Computational Methods for Control Applications, International Séries GAKUTO (2002). | Zbl 1082.93584

[14] V.G. Maz'Ya and S.A. Nazarov, Asymptotic behavior of energy integrals under small perturbations of the boundary near corner and conic points. Trudy Moskov. Mat. Obshch. 50 (1987) 79-129, 261. | Zbl 0668.35027

[15] V.G. Maz'Ya, S.A. Nazarov and B.A. Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Birkhäuser, Berlin (2000).

[16] S.A. Nazarov and M.V. Olyushin, Perturbation of the eigenvalues of the Neumann problem due to the variation of the domain boundary. Algebra i Analiz 5 (1993) 169-188. | Zbl 0827.35086

[17] S.A. Nazarov and J. Sokołowski, Asymptotic analysis of shape functionals. J. Math. Pures Appl. 82 (2003) 125-196. | Zbl 1031.35020

[18] S. Tordeux and G. Vial, Matching of asymptotic expansions and multiscale expansion for the rounded corner problem. SAM Research Report, ETH, Zürich (2006).