0.5 setgray0 0.5 setgray1 An adaptive PML technique for time-harmonic scattering problems Following a paper by Zhiming Chen and Xuezhe Liu Manuel Largo An adaptive PML technique for time-harmonic scattering problems – p. 1/51
Overview Introduction, Hankel functions An adaptive PML technique for time-harmonic scattering problems – p. 2/51
Overview Introduction, Hankel functions PML formulation An adaptive PML technique for time-harmonic scattering problems – p. 2/51
Overview Introduction, Hankel functions PML formulation Finite Elements and the Main Theorem An adaptive PML technique for time-harmonic scattering problems – p. 2/51
Overview Introduction, Hankel functions PML formulation Finite Elements and the Main Theorem Implementation and Examples An adaptive PML technique for time-harmonic scattering problems – p. 2/51
First Part INTRODUCTION AND HANKEL FUNCTIONS An adaptive PML technique for time-harmonic scattering problems – p. 3/51
Task We want to show how we can adapt finite element mesh size. An adaptive PML technique for time-harmonic scattering problems – p. 4/51
Task We want to show how we can adapt finite element mesh size. To do so, we need an a posteriori error estimate to control the error we make when discretizing space. An adaptive PML technique for time-harmonic scattering problems – p. 4/51
Task We want to show how we can adapt finite element mesh size. To do so, we need an a posteriori error estimate to control the error we make when discretizing space. We extend the idea of using a posteriori error estimates to determine the PML parameters and propose an adaptive PML technique for solving the Helmholtz-type scattering problem. An adaptive PML technique for time-harmonic scattering problems – p. 4/51
Task We want to show how we can adapt finite element mesh size. To do so, we need an a posteriori error estimate to control the error we make when discretizing space. We extend the idea of using a posteriori error estimates to determine the PML parameters and propose an adaptive PML technique for solving the Helmholtz-type scattering problem. We will first introduce and prove some error estimates, later construct an algorithm to adapt mesh size with a posteriori error control. An adaptive PML technique for time-harmonic scattering problems – p. 4/51
Scattering problem So, lets derive a PML technique for solving Helmholtz-type scattering problems with perfectly conducting boundary. An adaptive PML technique for time-harmonic scattering problems – p. 5/51
Scattering problem So, lets derive a PML technique for solving Helmholtz-type scattering problems with perfectly conducting boundary. Let D ∈ R 2 denote the bounded domain (scatterer) with boundary Γ D , g ∈ H − 1 / 2 (Γ D ) determined by the incoming wave, n the unit outer normal to Γ D . An adaptive PML technique for time-harmonic scattering problems – p. 5/51
Scattering problem So, lets derive a PML technique for solving Helmholtz-type scattering problems with perfectly conducting boundary. Let D ∈ R 2 denote the bounded domain (scatterer) with boundary Γ D , g ∈ H − 1 / 2 (Γ D ) determined by the incoming wave, n the unit outer normal to Γ D . Helmholtz-type scattering problem (constant k ): in R 2 \ ¯ ∆ u + k 2 u = 0 D ∂u = − g on Γ D ∂ n √ r � ∂u � ∂r − i ku → 0 as r = | x | → ∞ An adaptive PML technique for time-harmonic scattering problems – p. 5/51
Hankel functions First, consider the Bessel equation for functions of order ν : z 2 d 2 y dz 2 + z dy dz + ( z 2 − ν 2 ) y = 0 , ν ∈ C . An adaptive PML technique for time-harmonic scattering problems – p. 6/51
Hankel functions First, consider the Bessel equation for functions of order ν : z 2 d 2 y dz 2 + z dy dz + ( z 2 − ν 2 ) y = 0 , ν ∈ C . The so called Bessel function of the first kind J ν ( z ) is defined as the solution to the Bessel differential equation with non singular values at the origin. An adaptive PML technique for time-harmonic scattering problems – p. 6/51
Hankel functions First, consider the Bessel equation for functions of order ν : z 2 d 2 y dz 2 + z dy dz + ( z 2 − ν 2 ) y = 0 , ν ∈ C . The so called Bessel function of the first kind J ν ( z ) is defined as the solution to the Bessel differential equation with non singular values at the origin. An adaptive PML technique for time-harmonic scattering problems – p. 6/51
Hankel functions (cont) The so called Bessel function of the second kind Y ν ( z ) is defined as the solution to the Bessel differential equation with singular values at the origin. An adaptive PML technique for time-harmonic scattering problems – p. 7/51
Hankel functions (cont) The so called Bessel function of the second kind Y ν ( z ) is defined as the solution to the Bessel differential equation with singular values at the origin. An adaptive PML technique for time-harmonic scattering problems – p. 7/51
Hankel functions (cont) We introduce now the Hankel function of the first kind and order ν H (1) ν ( z ) , z ∈ C , and the Hankel function of the second kind and order ν H (2) ν ( z ) , z ∈ C , are defined by H (1) ν ( z ) ≡ J ν ( z ) + i Y ν ( z ) , H (2) ≡ J ν ( z ) − i Y ν ( z ) . ν ( z ) An adaptive PML technique for time-harmonic scattering problems – p. 8/51
Hankel functions (cont) We introduce now the Hankel function of the first kind and order ν H (1) ν ( z ) , z ∈ C , and the Hankel function of the second kind and order ν H (2) ν ( z ) , z ∈ C , are defined by H (1) ν ( z ) ≡ J ν ( z ) + i Y ν ( z ) , H (2) ≡ J ν ( z ) − i Y ν ( z ) . ν ( z ) Asymptotic behaviour: � 2 πz e i ( z − 1 2 νπ − 1 H (1) 4 π ) , ν ( z ) ∼ � 2 πz e − i ( z − 1 2 νπ − 1 H (2) 4 π ) . ν ( z ) ∼ An adaptive PML technique for time-harmonic scattering problems – p. 8/51
Hankel functions, H (1) 0 An adaptive PML technique for time-harmonic scattering problems – p. 9/51
Hankel functions, H (1) 1 An adaptive PML technique for time-harmonic scattering problems – p. 10/51
Hankel functions, H (1) − 1 An adaptive PML technique for time-harmonic scattering problems – p. 11/51
Lemma 1 Lemma 1 : For any ν ∈ R , z ∈ C ++ = { z ∈ C : ℑ ( z ) ≥ 0 , ℜ ( z ) ≥ 0 } , and Θ ∈ R such that 0 < Θ ≤ | z | , we have r 1 − Θ2 | z | 2 | H (1) −ℑ ( z ) | H (1) ν ( z ) | ≤ e ν (Θ) | An adaptive PML technique for time-harmonic scattering problems – p. 12/51
Lemma 1 Lemma 1 : For any ν ∈ R , z ∈ C ++ = { z ∈ C : ℑ ( z ) ≥ 0 , ℜ ( z ) ≥ 0 } , and Θ ∈ R such that 0 < Θ ≤ | z | , we have r 1 − Θ2 | z | 2 | H (1) −ℑ ( z ) | H (1) ν ( z ) | ≤ e ν (Θ) | An adaptive PML technique for time-harmonic scattering problems – p. 12/51
Lemma 1 (cont.) An adaptive PML technique for time-harmonic scattering problems – p. 13/51
Second Part PML FORMULATION An adaptive PML technique for time-harmonic scattering problems – p. 14/51
Setup Let the scatterer D be contained in the interior of the circle B R = { x ∈ R 2 : | x | < R } , and Ω R = B R \ ¯ D . We now surround the domain Ω R with a PML layer Ω PML = { x ∈ R 2 : R < | x | < ρ } . An adaptive PML technique for time-harmonic scattering problems – p. 15/51
The PML formulation Look at the domain R 2 \ ¯ B R . The solution u of the scattering problem can be written under the polar coordinates as follows: � 2 π H (1) n ( kr ) u n = 1 � u n e i nθ , u ( R, θ ) e − i nθ dθ. u ( r, θ ) = ˆ ˆ H (1) 2 π n ( kR ) 0 n ∈ Z H (1) denotes the just discussed Hankel function of the first kind and order n n . It can be shown that this series converges uniformly for r > R . An adaptive PML technique for time-harmonic scattering problems – p. 16/51
Dirichlet-to-Neumann operator We now introduce the so called Dirichlet-to-Neumann operator T : H 1 / 2 (Γ R ) → H − 1 / 2 (Γ R ) , where Γ R = ∂B R . It is definied as follows: for any f ∈ H 1 / 2 (Γ R ) , � 2 π k H (1) ′ ( kR ) f n = 1 n ˆ ˆ � f n e i nθ , fe − i nθ dθ. Tf = H (1) 2 π n ( kR ) 0 n ∈ Z An adaptive PML technique for time-harmonic scattering problems – p. 17/51
Dirichlet-to-Neumann operator We now introduce the so called Dirichlet-to-Neumann operator T : H 1 / 2 (Γ R ) → H − 1 / 2 (Γ R ) , where Γ R = ∂B R . It is definied as follows: for any f ∈ H 1 / 2 (Γ R ) , � 2 π k H (1) ′ ( kR ) f n = 1 n ˆ ˆ � f n e i nθ , fe − i nθ dθ. Tf = H (1) 2 π n ( kR ) 0 n ∈ Z Looking at the representation of the solution u in polar coordinates: � 2 π H (1) n ( kr ) u n = 1 � u n e i nθ , u ( R, θ ) e − i nθ dθ, u ( r, θ ) = ˆ ˆ H (1) 2 π n ( kR ) 0 n ∈ Z it is obvious that it satisfies ∂u Γ R = Tu. ∂ n An adaptive PML technique for time-harmonic scattering problems – p. 17/51
Reformulation Let a : H 1 (Ω R ) × H 1 (Ω R ) → C be the sesquilinear form � ∇ ϕ · ∇ ¯ ψ − k 2 ϕ ¯ � � dx − � Tϕ, ψ � Γ R . a ( ϕ, ψ ) = ψ Ω R An adaptive PML technique for time-harmonic scattering problems – p. 18/51
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