A multiscale correction method for local singular perturbations of the boundary.

*(English)*Zbl 1129.65084The 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.

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.

Reviewer: Petr Necesal (Plzen)

##### 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 |

##### Keywords:

multiscale asymptotic analysis; shape optimization; patch of elements; Poisson equation; numerical examples; finite elements; asymptotic expansion; Laplace equation##### References:

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