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A multiscale correction method for local singular perturbations of the boundary. (English) Zbl 1129.65084
The authors deal with an elliptic partial differential equation in a domain with a small local boundary perturbation. The complete asymptotic expansion of its solution with respect to the size of the perturbing pattern is given. Moreover, the authors derive the variation of the associated energy (topological derivative) and propose a numerical procedure for the approximation of the solution.
Let \(\Omega_{0}\) be an open bounded subset of \(\mathbb R^{2}\) with smooth boundary containing the origin \(O\). Moreover, let the boundary \(\partial \Omega_{0}\) coincide with a straight line near the origin, precisely for \(| x| < r^{*}\). On the other hand, let \(H_{\infty}\) denote an infinite domain of \(\mathbb R^{2}\), which coincides with the upper half-plane at infinity, precisely for \(| x| > \mathbb R^{*}\). The perturbed domain \(\Omega_{\varepsilon}\) is defined for small \(\varepsilon\) by
\[ \Omega_{\varepsilon} = \{ x\in\Omega_{0} : | x| > \varepsilon\mathbb R^{*} \} \cup \{ x\in\varepsilon H_{\infty}: | x| < r^{*} \} \]
(note that there is no assumption of inclusion of \(\Omega_{\varepsilon}\) into the original one or conversely). Let \(u_{\varepsilon}\in H^{1}(\Omega_{\varepsilon})\) be the solution of the equation \(-\Delta u_{\varepsilon} = f\) in \(\Omega_{\varepsilon}\), where \(f\) is some function in \(L^{2}(\Omega_{0})\) vanishing in a neighborhood of the origin. Dirichlet boundary conditions are given on \(\Gamma_{D}\subset \partial \Omega_{\varepsilon}\) (which does not reach the origin) and the Neumann boundary conditions elsewhere.
The paper is organized as follows. The first section is devoted to the multiscale asymptotic method in order to obtain the full asymptotic expansion of the state function in the case of a straight boundary near the origin. These results are then extended to a curved case. In the next section, the leading terms in the asymptotical description of the energy functional are derived. The last part of the paper is devoted to the numerical method using patch of elements near the perturbation. Finally, a numerical validation of the authors theoretical results is given in the studied model case of the Laplace equation.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35B25 Singular perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
49Q10 Optimization of shapes other than minimal surfaces
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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References:
[1] G. Allaire , Shape optimization by the homogenization method , Applied Mathematical Sciences 146. Springer-Verlag, New York ( 2002 ). MR 1859696 | Zbl 0990.35001 · 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 · Zbl 05147337
[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 · 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 . · Zbl 1167.35340 · eudml:209340
[6] B. Engquist and A. Majda , Absorbing boundary conditions for the numerical simulation of waves . Math. Comp. 31 ( 1977 ) 629 - 651 . Zbl 0367.65051 · Zbl 0367.65051 · doi:10.2307/2005997
[7] D. Givoli , Nonreflecting boundary conditions . J. Comput. Phys. 94 ( 1991 ) 1 - 29 . Zbl 0731.65109 · Zbl 0731.65109 · doi:10.1016/0021-9991(91)90135-8
[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 · 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 · 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 · 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 · Zbl 0656.76008 · doi:10.1137/0725044
[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 · Zbl 1050.49028
[13] M. Masmoudi , The Topological Asymptotic , in Computational Methods for Control Applications, International Séries GAKUTO ( 2002 ). Zbl 1082.93584 · 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 · 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 · 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 · Zbl 1031.35020 · doi:10.1016/S0021-7824(03)00004-7
[18] S. Tordeux and G. Vial , Matching of asymptotic expansions and multiscale expansion for the rounded corner problem . SAM Research Report, ETH, Zürich ( 2006 ). · Zbl 1109.35013
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