References Example 1 Example 2 Example 3 Example 4 Profiles Transformation of corner singularities in presence of small or large parameters Monique Dauge IRMAR, Universit´ e de Rennes 1, FRANCE Analysis and Numerics of Acoustic and Electromagnetic Problems October 17-22, 2016, Linz, Austria RICAM Special Semester on Computational Methods in Science and Engineering http://perso.univ-rennes1.fr/monique.dauge 1/23
References Example 1 Example 2 Example 3 Example 4 Profiles Outline 1 Joint works and discussions Example 1: Self-similar perturbations of a 2D corner 2 3 Example 2: Neumann-Robin boundary conditions 4 Example 3: Thin layers Example 4: Conducting material in 2D eddy current formulation 5 6 Profiles and corner layers 1/23
References Example 1 Example 2 Example 3 Example 4 Profiles Outline Joint works and discussions 1 Example 1: Self-similar perturbations of a 2D corner 2 3 Example 2: Neumann-Robin boundary conditions 4 Example 3: Thin layers 5 Example 4: Conducting material in 2D eddy current formulation 6 Profiles and corner layers Two strategies Example 1: Self-similar perturbations of a 2D corner Example 2: Neumann-Robin Conclusion 1/23
References Example 1 Example 2 Example 3 Example 4 Profiles Joint works Example 2 M. C OSTABEL , M. D AUGE , A singularly perturbed mixed boundary value problem, Comm. Partial Differential Equations 21, 11 & 12 (1996), 1919–1949. Example 3 G. C ALOZ , M. C OSTABEL , M. D AUGE , G. V IAL Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer Asymptotic Analysis 50 (1/2) (2006), 121–173. Example 1 M. D AUGE , S. T ORDEUX , AND G. V IAL Selfsimilar perturbation near a corner: matching versus multiscale expansions for a model problem, Around the research of Vladimir Maz’ya. II, vol. 12 of Int. Math. Ser. Springer (2010), 95–134. M. D AUGE P. D ULAR , L. K R ¨ AHENB ¨ UHL , V. P´ ERON , R. P ERRUSSEL , C. Example 4 P OIGNARD Corner asymptotics of the magnetic potential in the eddy-current model Mathematical Methods in the Applied Sciences 37, 13 (2014) 1924–1955 Example 1 M. C OSTABEL , M. D ALLA R IVA , M. D AUGE , P. M USOLINO Converging expansions for Lipschitz self-similar perforations of a plane sector, In preparation , (2016). 2D eddy current formulation for a conductor surrounded by a dielectric medium Recent discussions with R. H IPTMAIR , R. C ASAGRANDE , K. S CHMIDT , A. S EMIN 1/23
References Example 1 Example 2 Example 3 Example 4 Profiles Outline Joint works and discussions 1 Example 1: Self-similar perturbations of a 2D corner 2 3 Example 2: Neumann-Robin boundary conditions 4 Example 3: Thin layers 5 Example 4: Conducting material in 2D eddy current formulation 6 Profiles and corner layers Two strategies Example 1: Self-similar perturbations of a 2D corner Example 2: Neumann-Robin Conclusion 2/23
References Example 1 Example 2 Example 3 Example 4 Profiles Example 1: Self-similar perturbations of a 2D corner See references Let Γ be the infinite plane sector 1 Γ = { x = ( x 1 , x 2 ) ∈ R 2 , r > 0 , θ ∈ ( 0 , ω ) } with polar coordinates ( r , θ ) and the opening ω ∈ ( 0 , π ) ∪ ( π, 2 π ) . Bounded un-perturbed domain Ω ≡ Ω 0 , with Ω ⊂ Γ . Assume 2 Ω bounded connected with curvilinear polygonal boundary, ∃ r 0 > 0 such that B ( 0 , r 0 ) ∩ Ω = B ( 0 , r 0 ) ∩ Γ Bounded perturbation pattern P, with P ⊂ Γ . 3 P connected with curvilinear polygonal boundary, ∃ R 0 > 0 such that B ( 0 , R 0 ) ∁ ∩ P = B ( 0 , R 0 ) ∁ ∩ Γ Family of perturbed domains, with ε ∈ ( 0 , ε 0 ) 4 Ω ε = Ω ∩ ε P 2/23
References Example 1 Example 2 Example 3 Example 4 Profiles Dirichlet Problems for Example 1 Simplifying assumption f ∈ L 2 (Ω) with support outside a ball B ( 0 , r 1 ) General assumption f ∈ L 2 (Ω) analytic or C ∞ inside a ball B ( 0 , r 1 ) We choose the simplifying assumptions for the talk because A zero Taylor expansion of f at the corners allows to discard polynomials from corner asymptotics, so to avoid log terms and convergence issues. The family of problems under consideration is the family of Dirichlet problems for ε ∈ [ 0 , ε 0 ) � − ∆ u ε = f in Ω ε ( P ε ) u ε = 0 on ∂ Ω ε . Unique solution in u ε ∈ H 1 0 (Ω ε ) 3/23
References Example 1 Example 2 Example 3 Example 4 Profiles Two particular cases of Example 1 Question Behavior of u ε close to 0 (in the region B ( 0 , r 0 ) ∩ Ω ε ) as ε → 0. Let us consider two examples 1a A rounded corner: The boundary of P is smooth! 1b A cracked corner: The boundary of P has a crack Σ abutting at 0 (P = Γ \ Σ , with a straight segment Σ ). Set α = π ω In both examples, Ω = Ω 0 has a corner of opening ω at 0. Therefore [Kondrat’ev 1967] u 0 has an expansion as K γ k r k α sin k αθ, u 0 = u K � u K 0 , reg ∈ H m + 1 (Ω ∩ B ( 0 , r 0 )) , m = [( K + 1 ) α ] 0 , reg + k = 1 4/23
References Example 1 Example 2 Example 3 Example 4 Profiles Paradoxes on singularities for Example 1 On the one hand u 0 = γ 1 r α sin αθ + u 0 , reg , u 0 , reg ∈ H 2 (Ω ∩ B ( 0 , r 0 )) On the other hand 1a Rounded corner: u ε is smooth in Ω ε ∩ B ( 0 , r 0 ) 1 2 sin θ 1b A cracked corner: The first singularity of u ε behaves like r 2 Thus 1a u ε has no singularity but u 0 has one 1b u ε has a stronger singularity than u 0 How can the singularity r α sin αθ associated with ε = 0 1a Disappear as soon as ε > 0 1b Transform into a stronger singularity as soon as ε > 0 ? 5/23
References Example 1 Example 2 Example 3 Example 4 Profiles Outline Joint works and discussions 1 Example 1: Self-similar perturbations of a 2D corner 2 3 Example 2: Neumann-Robin boundary conditions 4 Example 3: Thin layers 5 Example 4: Conducting material in 2D eddy current formulation 6 Profiles and corner layers Two strategies Example 1: Self-similar perturbations of a 2D corner Example 2: Neumann-Robin Conclusion 6/23
References Example 1 Example 2 Example 3 Example 4 Profiles Example 2: Neumann-Robin boundary conditions See references Take a polygon Ω with straight sides Choose a side, denoted by Σ R . Denote ∂ Ω \ Σ R by Σ N Assume for simplicity that the openings of Ω at the two ends of Σ R are the same, say ω Assume for simplicity that π ω is not a rational number Assume for simplicity that f ∈ L 2 (Ω) with support outside neighborhoods of the ends of Σ R Then the Neumann-Robin problem is for ε ∈ [ 0 , ε 0 ) − ∆ u ε = f in Ω ( P ε ) ∂ n u ε = 0 on Σ N ε∂ n u ε + u ε = 0 on Σ R When ε = 0, we obtain a mixed Neumann-Dirichlet problem − ∆ u 0 = f in Ω ( P 0 ) ∂ n u 0 = 0 on Σ N u 0 = 0 on Σ R 6/23
References Example 1 Example 2 Example 3 Example 4 Profiles Example 2: Variational formulations and singularities Variational spaces: V ε = H 1 (Ω) , ε > 0 , V 0 = { u ∈ H 1 (Ω) , � and u Σ R = 0 } . � Variational formulations ( P ε ) � � � ∇ u ε · ∇ v d x + 1 u ε ∈ H 1 (Ω) , ∀ v ∈ H 1 (Ω) , u ε v d σ = f v d x ε Ω Σ R Ω � � ( P 0 ) u 0 ∈ V 0 , ∀ v ∈ V 0 , ∇ u 0 · ∇ v d x = f v d x Ω Ω Singularities of u 0 (for a Neumann-Dirichlet corner) recall that α = π ω r α/ 2 cos 1 r 3 α/ 2 cos 3 r 5 α/ 2 cos 5 2 αθ, 2 αθ, 2 αθ, . . . Singularities of u ε (for a Neumann-Robin corner): r α cos αθ + ε − 1 γ 1 r α + 1 cos ( α + 1 ) θ + ε − 2 γ 2 r α + 2 cos ( α + 2 ) θ + . . . r 2 α cos 2 αθ + ε − 1 γ ′ 1 r 2 α + 1 cos ( 2 α + 1 ) θ + ε − 2 γ ′ 2 r 2 α + 2 cos ( 2 α + 2 ) θ + . . . i.e. Neumann singularity plus successive shadows. Negative powers of ε 7/23
References Example 1 Example 2 Example 3 Example 4 Profiles Outline Joint works and discussions 1 Example 1: Self-similar perturbations of a 2D corner 2 3 Example 2: Neumann-Robin boundary conditions 4 Example 3: Thin layers 5 Example 4: Conducting material in 2D eddy current formulation 6 Profiles and corner layers Two strategies Example 1: Self-similar perturbations of a 2D corner Example 2: Neumann-Robin Conclusion 8/23
References Example 1 Example 2 Example 3 Example 4 Profiles Example 3: Thin layer See references Take a polygon Ω with straight sides Consider the union Σ of two consecutive sides σ 1 and σ 2 . Let O = σ 1 ∩ σ 2 be the corner inside Σ . Assume for simplicity that the opening ω at O is such that π ω �∈ Q Assume for simplicity that the openings of Ω at the two exterior corners c 1 and c 2 in Σ equal π 2 Define the layer L ε of width ε as the polygon formed by Σ , the segments σ ℓ,ε , ℓ = 1 , 2, parallel to σ ℓ at the distance ε outside Ω , and two perpendicular segments of length ε issued from c 1 and c 2 . Assume for simplicity that f ∈ L 2 (Ω) with support outside a neighb. of O Set Ω ε = Ω ∪ Σ ∪ L ε and consider the transmission problem, with a positive number β � = 1 � � � u ε ∈ H 1 0 (Ω ε ) , ∀ v ∈ H 1 0 (Ω ε ) , ∇ u ε · ∇ v d x + β ∇ u ε · ∇ v d x = f v d x . Ω L ε Ω Thus u 0 is solution of the Dirichlet pb on Ω = Ω 0 . 8/23
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