<figure><img src="image_1.png" /></figure>

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# Matching of asymptotic expansions for the wave propagation in media with thin slot

Sébastien Tordeux, Patrick Joly

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Sébastien Tordeux, Patrick Joly. Matching of asymptotic expansions for the wave propagation in media with thin slot. AG Analysis und Numerik, 2005, Baal, Switzerland. inria-00528070

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Submitted on 21 Oct 2010

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Matching of asymptotic expansions
for the wave propagation in media
with thin slot

Sébastien Tordeux and Patrick Joly

AG Analysis und Numerik, January 2005

INRIA-Rocquencourt-Projet POEMS

ETH-SAM
A typical application

How can we study the scattering in media with **thin slot**?

<figure><img src="image_2.png" /></figure>

A physical problem with two **caractéristical** lengths

$$ \begin{cases} \text{The wavelength } \lambda \\ \text{The width of the slot } \varepsilon \end{cases} $$
A typical application

How can we study the scattering in media with **thin slot**?

<figure><img src="image_3.png" /></figure>

An **asymptotic** case:

$$ \epsilon \ll \lambda $$
# The numerical difficulty

A **mesh step** smaller than $\varepsilon$

<figure><img src="image_4.png" /></figure>

This leads to **costly** computations
# Some references

- Thin slot:
  Harrington, Auckland (1980), Tatout (1996).

- Finite differences:
  Taflove (1995).

- Thin plates and junction theory,...
  Ciarlet, Le Dret, Dauge-Costabel.

- Matching of asymptotic expansions:
  McIver, Rawlins (1993), Il'in (1992).

- multiscale analysis
  Maz'ya, Nazarov, Plamenevskii (1991)
  Oleinik, Shamaev, Yosifian (1992)
# A simple problem

<figure><img src="image_5.png" /></figure>

Scalar wave equation:

$$ \frac{\partial^2 p^\epsilon}{\partial t^2} - \Delta p^\epsilon = f $$

Harmonic solution:

$$ p^{\epsilon}(x, y, t) = \exp(-i\omega t) u^{\epsilon}(x, y) $$

Helmholtz Equation:

$$ \Delta u^{\epsilon} + \omega^2 u^{\epsilon} = -f \quad \text{in } \Omega^{\epsilon} $$
# A simple problem

<figure><img src="image_6.png" /></figure>

**Outgoing solution at infinity:**

$$ \frac{\partial u^{\epsilon}}{\partial n} - i\omega u^{\epsilon} \le \frac{C}{r^2}, \quad \text{for } r \text{ large,} $$

**Neumann limit condition**
(rigid wall)

$$ \frac{\partial u^{\epsilon}}{\partial n} = 0 \quad \text{on } \partial\Omega^{\epsilon} $$
# A simple problem

<figure><img src="image_7.png" /></figure>

**Outgoing solution at infinity:**

$$ \frac{\partial u^{\epsilon}}{\partial n} - i\omega u^{\epsilon} \le \frac{C}{r^2}, \quad \text{for } r \text{ large,} $$

**Neumann limit condition**
(rigid wall)

$$ \frac{\partial u^{\epsilon}}{\partial n} = 0 \quad \text{on } \partial\Omega^{\epsilon} $$

With the **Dirichlet** limit condition, the transmission inside the slot is **negligible** ($o(\epsilon^\infty)$).
# A numerical computation

<figure><img src="image_8.png" /></figure>

Numerical computation done with the **high order finite elements code** of (M. Duruflé, INRIA)
# A numerical computation

Neumann

<figure><img src="image_9.png" /></figure>

Dirichlet

<figure><img src="image_10.png" /></figure>

$$ \frac{\varepsilon}{\lambda} = 0.5 $$

<figure><img src="image_11.png" /></figure>
# A numerical computation

Neumann

<figure><img src="image_12.png" /></figure>

Dirichlet

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

$$ \frac{\varepsilon}{\lambda} = 0.2 $$
# A numerical computation

Neumann

<figure><img src="image_15.png" /></figure>

Dirichlet

<figure><img src="image_16.png" /></figure>

$$ \frac{\epsilon}{\lambda} = 0.1 $$

<figure><img src="image_17.png" /></figure>
# Objectives

<figure><img src="image_18.png" /></figure>

* Introduce **accurate** numerical methods
# Objectives

<figure><img src="image_19.png" /></figure>

* Introduce **accurate** numerical methods

<figure><img src="image_20.png" /></figure>

* We need an **intermediate** zone

<figure><img src="image_21.png" /></figure>
# Objectives

<figure><img src="image_22.png" /></figure>

* Introduce **accurate** numerical methods

<figure><img src="image_23.png" /></figure>

* We need an **intermediate zone**

<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /></figure>

* A technique **the matching of asymptotic expansions**

<figure><img src="image_26.png" /></figure>

* Define **new approximate models** to compute the solution.

<figure><img src="image_27.png" /></figure>

* Use effectively “universal” technique of numerical computation (mesh refinement).
Contributions to the match. of as. exp.

<figure><img src="image_28.png" /></figure>

A new presentation of the **matching principle** (not allways clear) postulated by the english school.
Contributions to the match. of as. exp.

* A new presentation of the **matching principle** (not allways clear) postulated by the english school.

* The **mathematical justification** of this technique.

* The proof are **inspirated** by the multiscale technique

* **Existence and unicity** of the terms of the expansions.

* Specific technique: **error estimates.**
Three zones

<figure><img src="image_29.png" /></figure>

● Far field (2D field)

- Near field (boundary layer)

● Slot field (1D field)
Three zones

<figure><img src="image_30.png" /></figure>

The asymptotic assumptions:

$$ \varepsilon \ll \eta_H(\varepsilon) \ll \lambda, $$

$$ \varepsilon \ll \eta_S(\varepsilon) \ll \lambda. $$

$$ \varepsilon \to 0 $$

$$ \eta(\varepsilon) \to 0 $$

$$ \eta(\varepsilon)/\varepsilon \to +\infty $$
Three zones

<figure><img src="image_31.png" /></figure>

The asymptotic assumptions:

$$ \varepsilon \ll \eta_H(\varepsilon) \ll \lambda, $$

$$ \varepsilon \ll \eta_S(\varepsilon) \ll \lambda. $$

$$ \varepsilon \to 0 $$

$$ \eta(\varepsilon) \to 0 $$

$$ \eta(\varepsilon)/\varepsilon \to +\infty $$
Three zones

<figure><img src="image_32.png" /></figure>

The asymptotic assumptions:

$$ \varepsilon \ll \eta_H(\varepsilon) \ll \lambda, $$

$$ \varepsilon \ll \eta_S(\varepsilon) \ll \lambda. $$

$$ \varepsilon \to 0 $$

$$ \eta(\varepsilon) \to 0 $$

$$ \eta(\varepsilon)/\varepsilon \to +\infty $$
Three zones

<figure><img src="image_33.png" /></figure>

Near field

The asymptotic assumptions:

$$ \varepsilon \ll \eta_H(\varepsilon) \ll \lambda, $$

$$ \varepsilon \ll \eta_S(\varepsilon) \ll \lambda. $$

$$ \varepsilon \to 0 $$

$$ \eta(\varepsilon) \to 0 $$

$$ \eta(\varepsilon)/\varepsilon \to +\infty $$
Three zones

<figure><img src="image_34.png" /></figure>

Slot field

The asymptotic assumptions:

$$ \varepsilon \ll \eta_H(\varepsilon) \ll \lambda, $$

$$ \varepsilon \ll \eta_S(\varepsilon) \ll \lambda. $$

$$ \varepsilon \to 0 $$

$$ \eta(\varepsilon) \to 0 $$

$$ \eta(\varepsilon)/\varepsilon \to +\infty $$
Three zones

<figure><img src="image_35.png" /></figure>

Far and near

The asymptotic assumptions:

$$ \varepsilon \ll \eta_H(\varepsilon) \ll \lambda, $$

$$ \varepsilon \ll \eta_S(\varepsilon) \ll \lambda. $$

$$ \varepsilon \to 0 $$

$$ \eta(\varepsilon) \to 0 $$

$$ \eta(\varepsilon)/\varepsilon \to +\infty $$
Three zones

<figure><img src="image_36.png" /></figure>

Slot and near

The asymptotic assumptions:

$$ \varepsilon \ll \eta_H(\varepsilon) \ll \lambda, $$

$$ \varepsilon \ll \eta_S(\varepsilon) \ll \lambda. $$

$$ \varepsilon \to 0 $$

$$ \eta(\varepsilon) \to 0 $$

$$ \eta(\varepsilon)/\varepsilon \to +\infty $$
# The different steps of the method

<figure><img src="image_37.png" /></figure>

* Derivate the asymptotic expansions:

<figure><img src="image_38.png" /></figure>

* Formal part

<figure><img src="image_39.png" /></figure>

* Several presentations are possible
# The different steps of the method

* Derivate the asymptotic expansions:

* Formal part

* Several presentations are possible

* Describe the asymptotic expansions

* Rigorous part

* Definition of the terms of the asymptotic expansions
# The different steps of the method

* Derivate the asymptotic expansions:

* Formal part

* Several presentations are possible

* Describe the asymptotic expansions

* Rigorous part

* Definition of the terms of the asymptotic expansions

* Mathematical validation of the asymptotic expansions

* Rigorous part

* Error estimates
# The different steps of the method

2. Derivate the asymptotic expansions:

* Formal part

* Several presentations are possible

1. Describe the asymptotic expansions

* Rigorous part

* Definition of the terms of the asymptotic expansions

3. Mathematical validation of the asymptotic expansions

* Rigorous part

* Error estimates
# Far field

Asymptotic context: $\varepsilon \ll \eta_H \ll \lambda$.

<figure><img src="image_40.png" /></figure>
# Far field

Asymptotic context: $\varepsilon \ll \eta_H \ll \lambda$.

<figure><img src="image_41.png" /></figure>

No normalization:

$$X = x,$$

$$Y = y.$$
# Far field

Asymptotic context: $\epsilon \ll \eta_H \ll \lambda$.

<figure><img src="image_42.png" /></figure>

No normalization:

$X = x,$

$Y = y.$
Far field

Asymptotic context: $\varepsilon \ll \eta_H \ll \lambda$.

<figure><img src="image_43.png" /></figure>

$$ u^{\varepsilon} = u^{0} + \sum_{i=1}^{+\infty} \sum_{k=0}^{i-1} \varepsilon^{i} (\log \varepsilon)^{k} u_{i}^{k} + o(\varepsilon^{\infty}), \quad \text{in } \Omega. $$
Far field

Asymptotic context: $\varepsilon \ll \eta_H \ll \lambda$.

<figure><img src="image_44.png" /></figure>

where the $u_i^k$ satisfy the **homogeneous Helmholtz** equation

$$ \Delta u_i^k + \omega^2 u_i^k = 0. $$
Slot field

<figure><img src="image_45.png" /></figure>
Slot field

<figure><img src="image_46.png" /></figure>

$$u^{\epsilon}(x, y) = U^{\epsilon}(x, \frac{y}{\epsilon})$$
# Slot field

<figure><img src="image_47.png" /></figure>

The **asymptotic context**: $\varepsilon \ll \eta_S \ll \lambda$.

The **normalization**: $X = x, Y = \frac{y}{\varepsilon}$.
# Slot field

<figure><img src="image_48.png" /></figure>

The **asymptotic context**: $\varepsilon \ll \eta_S \ll \lambda$.

The **normalization**: $X = x, Y = \frac{y}{\varepsilon}$.
# Slot field

<figure><img src="image_49.png" /></figure>

The **asymptotic context**: $\varepsilon \ll \eta_S \ll \lambda$.

The **normalization**: $X = x, Y = \frac{y}{\varepsilon}$.
Slot field

<figure><img src="image_50.png" /></figure>

$$u^{\varepsilon}(x, Y\varepsilon) = U^{\varepsilon}(x, Y) = \sum_{i=0}^{+\infty} \sum_{k=0}^{i} \varepsilon^i (\log \varepsilon)^k U_i^k(x, Y) + o(\varepsilon^{\infty}),$$
Slot field

<figure><img src="image_51.png" /></figure>

where the $U_i^k$ satisfy the **1D Helmholtz** equation:

$$ \frac{d^2 U_i^k}{dx^2} + \omega^2 U_i^k = 0 $$
Near field

<figure><img src="image_52.png" /></figure>
Near field

<figure><img src="image_53.png" /></figure>

$$u^{\epsilon}(x, y) = u_{p}^{\epsilon}\left(\frac{x}{\epsilon}, \frac{y}{\epsilon}\right)$$
# Near field

<figure><img src="image_54.png" /></figure>

The **Asymptotic context**: $\varepsilon \ll \eta_H \ll \lambda$, $\varepsilon \ll \eta_S \ll \lambda$.

The **normalization**: $X = \frac{x}{\varepsilon}$, $Y = \frac{y}{\varepsilon}$
Near field

<figure><img src="image_55.png" /></figure>

The **Asymptotic context**: $\varepsilon \ll \eta_H \ll \lambda$, $\varepsilon \ll \eta_S \ll \lambda$.

The **normalization**: $X = \frac{x}{\varepsilon}$, $Y = \frac{y}{\varepsilon}$
Near field

<figure><img src="image_56.png" /></figure>

The **Asymptotic context**: $\varepsilon \ll \eta_H \ll \lambda$, $\varepsilon \ll \eta_S \ll \lambda$.

The **normalization**: $X = \frac{x}{\varepsilon}$, $Y = \frac{y}{\varepsilon}$
Near field

<figure><img src="image_57.png" /></figure>

The **Asymptotic context**: $\varepsilon \ll \eta_H \ll \lambda$, $\varepsilon \ll \eta_S \ll \lambda$.

The **normalization**: $X = \frac{x}{\varepsilon}$, $Y = \frac{y}{\varepsilon}$
Near field

<figure><img src="image_58.png" /></figure>

$$u^{\varepsilon}(\varepsilon X, \varepsilon Y) = u_{p}^{\varepsilon}(X, Y) = \sum_{i=0}^{+\infty} \sum_{k=0}^{i} \varepsilon^{i} (\log \varepsilon)^{k} (u_{p})_{i}^{k}(X, Y) + o(\varepsilon^{\infty})$$
Near field

<figure><img src="image_59.png" /></figure>

where the $(u_p)_i^k$ satisfy the (in)-homogeneous Laplace
equation.

$$
\begin{cases}
\Delta(\boldsymbol{u}_p)_i^k = 0, & \text{if } i = k \text{ or } k+1, \\
\Delta(\boldsymbol{u}_p)_i^k = -\omega^2 (\boldsymbol{u}_p)_{i-2}^k, & \text{else.}
\end{cases}
$$
**Order 0 :** $\underline{u}^0, (\underline{u}_p)^0, \underline{U}^0$

<figure><img src="image_60.png" /></figure>

**Far field:**

$$ egin{cases} 
\begin{array}{l} 
\text{Find } \underline{u}^0 \in H_{\text{loc}}^1(\Omega) \text{ such that :} \\ 
-\Delta \underline{u}^0 - \omega^2 \underline{u}^0 = f, \quad \text{in } \Omega, \\ 
\frac{\partial \underline{u}^0}{\partial n} = 0, \quad \text{on } \partial\Omega, \\ 
\underline{u}^0 \text{ is outgoing.} 
\end{array} 
\end{cases} $$
Order 0 : $u^0, (u_p)_0^0, U_0^0$

<figure><img src="image_61.png" /></figure>

Near field:

$$ (u_p)_0^0 (X, Y) = u^0(A), \quad \text{in } \mathcal{B}_1. $$
Order 0 : $u^0, (u_p)_0^0, U_0^0$

$O_1$

Slot field:

$U_0^0(x, Y) = u^0(A) \exp(i\omega x)$, in $O_1$.
Order 1 : $\underline{u_1^0}$, $(u_p)_1^0$, $(u_p)_1^1$, $U_1^0$, $U_1^1$

<figure><img src="image_62.png" /></figure>

Approximation of the exact Solution:

$$u^{\epsilon} \approx u^{0} + \epsilon u_{1}^{0}$$
Order 1 : $\underline{u_1^0}$, $(u_p)_1^0$, $(u_p)_1^1$, $U_1^0$, $U_1^1$

<figure><img src="image_63.png" /></figure>

explicit form of $\underline{u_1^0}$

$$ \underline{u_1^0}(r, \theta) = -\frac{\omega}{2} u^0(A) H_0^{(1)}(\omega r). $$
Order 1 : $u_1^0, (u_p)_1^0, (u_p)_1^1, U_1^0, U_1^1$

<figure><img src="image_64.png" /></figure>

Approximation of the exact solution:

$$ \left\{ \begin{array}{l} u^{\epsilon}(\epsilon X, \epsilon Y) = u_{p}^{\epsilon}(X, Y), \\ u_{p}^{\epsilon} \approx (u_{p})_{0}^{0} + \epsilon (u_{p})_{1}^{0} + \epsilon \log \epsilon (u_{p})_{1}^{1}. \end{array} \right. $$
Order 1 : $u_1^0, (u_p)_1^0, (u_p)_1^1, U_1^0, U_1^1$

<figure><img src="image_65.png" /></figure>

Near field:

Find $(u_p)_1^0 \in H_{loc}^1(\mathcal{B}_1)$ such that:

$$ egin{cases} 
\Delta(\color{red}{u_p})_1^0 = 0, & \text{in } \mathcal{B}_1 \\ 
\displaystyle\frac{\partial(\color{red}{u_p})_1^0}{\partial n} = 0, & \text{on } \partial\mathcal{B}_1. 
\end{cases} $$
Order 1 : $u_1^0, (u_p)_1^0, (u_p)_1^1, U_1^0, U_1^1$

<figure><img src="image_66.png" /></figure>

Behavior at infinity in the half-space:

$$ (u_p)_1^0(\rho, \theta) - \frac{\partial u^0}{\partial y}(A) \rho \cos\theta + \frac{\omega}{2} u^0(A) \left[ 1 + \frac{2i}{\pi} (\log \rho + \gamma) \right] = O\left(\frac{1}{\rho}\right). $$
Order 1 : $u_1^0, (u_p)_1^0, (u_p)_1^1, U_1^0, U_1^1$

<figure><img src="image_67.png" /></figure>

**Behavior at infinity in the half-space:**

$$ (u_p)_1^0(\rho, \theta) - \frac{\partial u^0}{\partial y}(A) \rho \cos \theta + \frac{\omega}{2} u^0(A) \left[ 1 + \frac{2i}{\pi} (\log \rho + \gamma) \right] = O\left(\frac{1}{\rho}\right). $$

**Behavior at infinity in the slot:**

$$ (u_p)_1^0 (X, Y) - i \omega u^0(A) X = O(1). $$
Order 1 : $u_1^0, (u_p)_1^0, (u_p)_1^1, U_1^0, U_1^1$

<figure><img src="image_68.png" /></figure>

$$ (u_p)_1^1 = -\frac{\dot{i}\omega}{\pi} u^0(A) $$
Order 1 : $u_1^0, (u_p)_1^0, (u_p)_1^1, \underline{U_1^0}, \underline{U_1^1}$

<figure><img src="image_69.png" /></figure>

Approximation of the exact solution:

$$ \begin{cases} u^{\epsilon}(x, \epsilon Y) = U^{\epsilon}(x, Y), \\ U^{\epsilon} \approx U_{0}^{0} + \epsilon U_{1}^{0} + \epsilon \log \epsilon U_{1}^{1}. \end{cases} $$
Order 1 : $u_1^0, (u_p)_1^0, (u_p)_1^1, \underline{U_1^0}, \underline{U_1^1}$

<figure><img src="image_70.png" /></figure>

The slot field:

$$ \mathbf{U}_1^0(x) = \int_0^1 \mathbf{U}_1^0(0, Y) \, dY \exp(i\omega x), $$
Order 1 : $u_1^0, (u_p)_1^0, (u_p)_1^1, U_1^0, U_1^1$

<figure><img src="image_71.png" /></figure>

The slot field:

$$U_1^1(x) = -\frac{\mathbf{i}\omega}{\pi} u^0(A) \exp(\mathbf{i}\omega x).$$
The far field of order $i > 1$

* The field $u_i^k$ are defined in the **half space**:

<figure><img src="image_72.png" /></figure>
The far field of order $i > 1$

* The field $u_i^k$ are defined in the **half space**:

<figure><img src="image_73.png" /></figure>

* The far fields $u_i^k$

* satisfy the **homogeneous Helmholtz** equation

* are **singular** at the neighborhood of the origin

* are outgoing at infinity
The far field of order $i > 1$

* The field $u_i^k$ are defined in the **half space**:

<figure><img src="image_74.png" /></figure>

* $u_i^k = \sum_{p=0}^{+\infty} a_p H_p^{(1)}(\omega r) \cos p\theta$
The far field of order $i > 1$

* The field $u_i^k$ are defined in the **half space**:

<figure><img src="image_75.png" /></figure>

* $u_i^k = \sum_{p=0}^{i-k-1} a_p H_p^{(1)}(\omega r) \cos p\theta$
The far field of order $i > 1$

* The field $u_i^k$ are defined in the **half space**:

<figure><img src="image_76.png" /></figure>

* $u_i^k = \sum_{p=0}^{i-k-1} a_p H_p^{(1)}(\omega r) \cos p\theta$

The $a_p$ are functions of **lower order** terms
The far field of order $i > 1$

* The field $u_i^k$ are defined in the **half space**:

<figure><img src="image_77.png" /></figure>

* $u_i^k = \sum_{p=0}^{i-k-1} a_p H_p^{(1)}(\omega r) \cos p\theta$

The $a_p$ are functions of **lower order** terms
# The far field of order $i > 1$

* The field $u_i^k$ are defined in the **half space**:

<figure><img src="image_78.png" /></figure>

* $u_i^k = \sum_{p=0}^{i-k-1} a_p H_p^{(1)}(\omega r) \cos p\theta$

The $a_p$ are functions of **lower order** terms
The near fields of order $i > 1$

* The $(u_p)_i^k(X, Y)$ are defined in the **canonical** domain:

<figure><img src="image_79.png" /></figure>
# The near fields of order $i > 1$

* The $(u_p)_i^k(X, Y)$ are defined in the **canonical** domain:

<figure><img src="image_80.png" /></figure>

* by **Laplace** equation:

$$ \Delta(\boldsymbol{u}_p)_i^k = 0, \quad (i = k \text{ ou } k + 1), $$

$$ \Delta(\boldsymbol{u}_p)_i^k = -\omega^2 (\boldsymbol{u}_p)_{i-2}^k, \quad (i \ge k + 2), $$
The near fields of order $i > 1$

* The $(u_p)_i^k(X, Y)$ are defined in the **canonical** domain:

<figure><img src="image_81.png" /></figure>

* by **Laplace** equation:

* by polynomial **growings** at infinity:

* The **growings** in the half space are functions of **far field of lower (or equal) order**

* The **growings** in the slot are functions of the slot fields **of lower order**
# The near fields of order $i > 1$

* The $(u_p)_i^k(X, Y)$ are defined in the **canonical** domain:

<figure><img src="image_82.png" /></figure>

* Proof of the **existence-unicity**:

* with truncature functions, we subtract the growing behavior at infinity of the $(u_p)_i^k$

* We use the “classical” **variational theory** (wheighted Sobolev spaces, Leroux, Hardy,...)
The slot field of order $i > 1$

* The $U_i^k$ are defined on the **canonical** domain:

<figure><img src="image_83.png" /></figure>
The slot field of order $i > 1$

* The $U_i^k$ are defined on the **canonical** domain:

<figure><img src="image_84.png" /></figure>

* The $U_i^k$ does not depend on $y$.
The slot field of order $i > 1$

* The $U_i^k$ are defined on the **canonical** domain:

<figure><img src="image_85.png" /></figure>

* The $U_i^k$ does not depend on $y$.

* $U_i^k(x) = \int_0^1 (u_p)_i^k(0, Y)dY \exp i\omega x$
# Some properties

We see that:

<figure><img src="image_86.png" /></figure>

More $i-k$ is **large** more $u_i^k$ is **singular** at the origin:

$r^{-p}$ terms, $p = 0, \dots, i-k-1$
# Some properties

We see that:

* More $i-k$ is **large** more $u_i^k$ is **singular** at the origin:

$r^{-p}$ terms, $p = 0, \dots, i-k-1$

* More $i-k$ is **large** more $(u_p)_i^k$ is **growing**:

$\rho^p$ terms, $p = 0, \dots, i-k$,
$X^p$ terms, $p = 0, \dots, i-k$,
# Some properties

We see that:

* More $i-k$ is **large** more $u_i^k$ is **singular** at the origin:

$r^{-p}$ terms, $p = 0, \dots, i-k-1$

* More $i-k$ is **large** more $(u_p)_i^k$ is **growing**:

$\rho^p$ terms, $p = 0, \dots, i-k,$
$X^p$ terms, $p = 0, \dots, i-k,$

* When the **order** $i$ grows, one has $O(\frac{i^2}{2}) (\times 3)$ terms to compute...
# Dependence diagram of the asymp. terms

<figure><img src="image_87.png" /></figure>

Any point corresponds to the 3 functions ($u_i^k, (u_p)_i^k, U_i^k$).
$$ \epsilon^i \log^k \epsilon $$

<figure><img src="image_88.png" /></figure>

Any point corresponds to the 3 functions ($u_i^k$, $(u_p)_i^k$, $U_i^k$).
<figure><img src="image_89.png" /></figure>

Any point corresponds to the three functions $(u_i^k, (u_p)_i^k, U_i^k)$.
# Devirvate the terms of the as. exp.

<figure><img src="image_90.png" /></figure>

We search for solutions of the form:

$$ \sum_{i \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} \varepsilon^i (\log \varepsilon)^k u_i^k \qquad \text{(far field)} $$

$$ \sum_{i \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} \varepsilon^i (\log \varepsilon)^k (u_p)_i^k \qquad \text{(near field)} $$

$$ \sum_{i \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} \varepsilon^i (\log \varepsilon)^k U_i^k \qquad \text{(slot field)} $$
# Devirvate the terms of the as. exp.

<figure><img src="image_91.png" /></figure>

We search for solutions of the form:

$$ \sum_{i \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} \varepsilon^i (\log \varepsilon)^k u_i^k \qquad \text{(far field)} $$

$$ \sum_{i \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} \varepsilon^i (\log \varepsilon)^k (u_p)_i^k \qquad \text{(near field)} $$

$$ \sum_{i \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} \varepsilon^i (\log \varepsilon)^k U_i^k \qquad \text{(slot field)} $$

<figure><img src="image_92.png" /></figure>

We **inject** the equations (Helmholtz, Neumann)
# Devirvate the terms of the as. exp.

<figure><img src="image_93.png" /></figure>

We search for solutions of the form:

$$ \sum_{i \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} \varepsilon^i (\log \varepsilon)^k u_i^k \qquad \text{(far field)} $$

$$ \sum_{i \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} \varepsilon^i (\log \varepsilon)^k (u_p)_i^k \qquad \text{(near field)} $$

$$ \sum_{i \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} \varepsilon^i (\log \varepsilon)^k U_i^k \qquad \text{(slot field)} $$

<figure><img src="image_94.png" /></figure>

We **inject** the equations (Helmholtz, Neumann)

<figure><img src="image_95.png" /></figure>

We obtain the **coupling conditions**: (**the difficulty**)
# Far-Near coupling

<figure><img src="image_96.png" /></figure>

In a **thick zone**:

$$ \epsilon \ll \eta_H \ll \lambda. $$

We write the coupling condition:

$$ u^{\epsilon}(\eta_H, \theta) = (u_p)^{\epsilon}(\frac{\eta_H}{\epsilon}, \theta). $$

$$ \begin{gathered} \sum_{i \in \mathbb{Z}}^{+\infty} \sum_{k \in \mathbb{Z}}^{\epsilon} (\log \epsilon)^k u_i^k(\eta_H, \theta) \approx \sum_{i \in \mathbb{Z}}^{+\infty} \sum_{k \in \mathbb{Z}}^{\epsilon} (\log \epsilon)^k (u_p)_i^k(\frac{\eta_H}{\epsilon}, \theta) \\ \eta_H \to 0 \qquad \frac{\eta_H}{\epsilon} \to +\infty \end{gathered} $$
Far-Near coupling

$$
\sum_{i \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} \varepsilon^i (\log \varepsilon)^k u_i^k(\eta_H, \theta) \approx \sum_{i \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} \varepsilon^i (\log \varepsilon)^k (u_p)_i^k(\frac{\eta_H}{\varepsilon}, \theta) \\
\eta_H \to 0 \qquad \frac{\eta_H}{\varepsilon} \to +\infty
$$

We **expand**

● the left serie according to $\eta_H$ near 0

- The right serie according to $\eta_H / \epsilon$ tending ot **infinity**

We **identify** all the terms of the two series.
# The conclusion of the coupling

<figure><img src="image_97.png" /></figure>

The far field-near field coupling:

The singular behavior of the far field is coupled with the none growing behavior of the near field at infinity.

<figure><img src="image_98.png" /></figure>
# The conclusion of the coupling

<figure><img src="image_99.png" /></figure>

The **far field-near field coupling:**

The **growing** behavior of the **near field at infinity** is coupled
with the **none singular** behavior of the **far field**.

<figure><img src="image_100.png" /></figure>
# The conclusion of the coupling

<figure><img src="image_101.png" /></figure>

The far field-near field coupling:

<figure><img src="image_102.png" /></figure>

The near field-slot field coupling:

The **growing** behavior of the **near field** is coupled with the
**none growing** behavior of the **slot field** (derivative values)

<figure><img src="image_103.png" /></figure>

<figure><img src="image_104.png" /></figure>
Mathematical analysis

<figure><img src="image_105.png" /></figure>

$$ \left\| \mathbf{u}^{\varepsilon} - \mathbf{u}^{0} - \sum_{i=1}^{n} \sum_{k=0}^{i-1} \varepsilon^{i} (\log \varepsilon)^{k} \mathbf{u}_{i}^{k} \right\|_{H^{1}(\Omega^{R,R'})} \leq C \varepsilon^{n+1} (\log \varepsilon)^{n} \|f\|_{L^{2}(\Omega)}. $$
Mathematical analysis

<figure><img src="image_106.png" /></figure>

<figure><img src="image_107.png" /></figure>

$$ \left\| \textcolor{red}{u^{\varepsilon}} - \textcolor{red}{u^0} - \sum_{i=1}^{n} \sum_{k=0}^{i-1} \varepsilon^i (\log \varepsilon)^k \textcolor{blue}{u_i^k} \right\|_{H^1(\Omega^{R,R'})} \leq C \varepsilon^{n+1} (\log \varepsilon)^n \|f\|_{L^2(\Omega)}. $$

$$ \left\| \textcolor{red}{u_p^{\varepsilon}} - \sum_{i=0}^{n} \sum_{k=0}^{i} \varepsilon^i (\log \varepsilon)^k (\textcolor{blue}{u_p})_i^k \right\|_{H^1(\mathcal{B}_1^{L,L'})} \leq C \varepsilon^{n+1} (\log \varepsilon)^{n+1} \|f\|_{L^2(\Omega)}. $$
Mathematical analysis

<figure><img src="image_108.png" /></figure>

<figure><img src="image_109.png" /></figure>

<figure><img src="image_110.png" /></figure>

$$ \left\| \textcolor{red}{u^{\varepsilon}} - \textcolor{red}{u^0} - \sum_{i=1}^{n} \sum_{k=0}^{i-1} \varepsilon^i (\log \varepsilon)^k \textcolor{red}{u_i^k} \right\|_{H^1(\Omega^{R,R'})} \leq C \varepsilon^{n+1} (\log \varepsilon)^n \|f\|_{L^2(\Omega)}. $$

$$ \left\| \textcolor{red}{u_p^{\varepsilon}} - \sum_{i=0}^{n} \sum_{k=0}^{i} \varepsilon^i (\log \varepsilon)^k (\textcolor{red}{u_p})_i^k \right\|_{H^1(\mathcal{B}_1^{L,L'})} \leq C \varepsilon^{n+1} (\log \varepsilon)^{n+1} \|f\|_{L^2(\Omega)}. $$

$$ \left\| \textcolor{red}{U^{\varepsilon}} - \sum_{i=0}^{n} \sum_{k=0}^{i} \varepsilon^i (\log \varepsilon)^k \textcolor{red}{U_i^k} \right\|_{H^1(\mathcal{O}_1^{l,l'})} \leq C \varepsilon^{n+1} (\log \varepsilon)^{n+1} \|f\|_{L^2(\Omega)}. $$
# Idea of the proof

We want to define an **approximation** $\tilde{u}_n^\varepsilon$ of the exact solution which **coincide** with:

* the **truncated** expansion of the **far field** away from the slot in the half space.

$$u_n^{H,\varepsilon}(x,y) = u^0(x,y) + \sum_{i=1}^{n} \sum_{k=0}^{i-1} \varepsilon^i (\log \varepsilon)^k u_i^k(x,y)$$

<figure><img src="image_111.png" /></figure>
# Idea of the proof

We want to define an **approximation** $\tilde{u}_n^\epsilon$ of the exact solution which **coincide** with:

* The **truncated** expansion of the **near field** in the neighbourhood of the end of the slot

$$u_n^{N, \epsilon}(x, y) = \sum_{i=0}^{n} \sum_{k=0}^{i} \epsilon^i (\log \epsilon)^k (u_p)_i^k \left(\frac{x}{\epsilon}, \frac{y}{\epsilon}\right)$$

<figure><img src="image_112.png" /></figure>
# Idea of the proof

We want to define an **approximation** $\tilde{u}_n^\varepsilon$ of the exact solution which **coincide** with:

* the **truncated** expansion of the **slot field** far away in the slot

$$u_n^{S,\varepsilon}(x,y) = \sum_{i=0}^{n} \sum_{k=0}^{i} \varepsilon^i (\log \varepsilon)^k U_i^k(x, \frac{y}{\varepsilon})$$
Idea of the proof

Introduce a partition of unity

$$ \tilde{u}_{n}^{\epsilon}(r, \theta) = \chi_{H}^{\epsilon} u_{n}^{H, \epsilon} + \chi_{N}^{\epsilon} u_{n}^{N, \epsilon} + \chi_{S}^{\epsilon} u_{n}^{S, \epsilon} $$

with

$$ \chi_{H}^{\epsilon} + \chi_{N}^{\epsilon} + \chi_{S}^{\epsilon} = 1. $$

<figure><img src="image_113.png" /></figure>
Idea of the proof

The error equation: $e_n^\epsilon = \tilde{u}_n^\epsilon - u^\epsilon$

$$
\left\{
\begin{array}{ll}
\Delta e_n^\epsilon + \omega^2 e_n^\epsilon = (\delta_N)_n^\epsilon + (\delta_{H-N})_n^\epsilon + (\delta_{S-N})_n^\epsilon, & \text{in } \Omega_\epsilon, \\
\\
\displaystyle\frac{\partial e_n^\epsilon}{\partial n} = 0, & \text{on } \partial\Omega_\epsilon, \\
\\
e_n^\epsilon \text{ is outgoing.}
\end{array}
\right.
$$

$(\delta_N)_n^\epsilon$ is related to the **approximation** of the **Helmholtz**
equation by the **near** field

<figure><img src="image_114.png" /></figure>
Idea of the proof

The error equation $e_n^\varepsilon = \tilde{u}_n^\varepsilon - u^\varepsilon$

$$
\left\{
\begin{array}{l}
\Delta e_n^\varepsilon + \omega^2 e_n^\varepsilon = (\delta_N)_n^\varepsilon + (\delta_{H-N})_n^\varepsilon + (\delta_{S-N})_n^\varepsilon, \quad \text{dans } \Omega_\varepsilon, \\
\\
\displaystyle\frac{\partial e_n^\varepsilon}{\partial n} = 0, \qquad \text{on } \partial\Omega_\varepsilon, \\
\\
e_n^\varepsilon \text{ is outgoing.}
\end{array}
\right.
$$

($\delta_{H-N}$)$_n^\epsilon$ is related to the **matching error** between the **far**
field ans the **near** field

<figure><img src="image_115.png" /></figure>
Idea of the proof

The error equation $e_n^\varepsilon = \tilde{u}_n^\varepsilon - u^\varepsilon$

$$
\left\{
\begin{array}{l}
\Delta e_n^\varepsilon + \omega^2 e_n^\varepsilon = (\delta_N)_n^\varepsilon + (\delta_{H-N})_n^\varepsilon + (\delta_{S-N})_n^\varepsilon, \quad \text{dans } \Omega_\varepsilon, \\
\\
\displaystyle\frac{\partial e_n^\varepsilon}{\partial n} = 0, \qquad \text{sur } \partial\Omega_\varepsilon, \\
\\
e_n^\varepsilon \text{ is outgoing.}
\end{array}
\right.
$$

($\delta_{S-N})_n^\epsilon$ is related to the **matching error** between the **slot**
field and the **near champ**

<figure><img src="image_116.png" /></figure>
Idea of the proof

The error equation: $e_n^\epsilon = \tilde{u}_n^\epsilon - u^\epsilon$

$$
\left\{
\begin{array}{ll}
\Delta e_n^\epsilon + \omega^2 e_n^\epsilon = (\delta_N)_n^\epsilon + (\delta_{H-N})_n^\epsilon + (\delta_{S-N})_n^\epsilon, & \text{in } \Omega_\epsilon, \\
\\
\displaystyle\frac{\partial e_n^\epsilon}{\partial n} = 0, & \text{on } \partial\Omega_\epsilon, \\
\\
e_n^\epsilon \text{ os outgoing.}
\end{array}
\right.
$$

classical asymptotic techniques:

● **Stability:** proof by **contradiction** (Helmholtz)

* **Consistency:** A little bit more difficult (study of the singularities and of the growings by separation of variable)
# Idea of the proof

## Global error estimates

$$ \left\{ \begin{array}{l} \|u^{\varepsilon} - \tilde{u}_{n}^{\varepsilon}\|_{H^1(\Omega_{\varepsilon}^{R,\delta})} \le C \left[ \left(\eta_H(\varepsilon)\right)^n + \left(\frac{\varepsilon}{\eta_H(\varepsilon)}\right)^n \right] \\ \qquad + C \left[ \left(\eta_S(\varepsilon)\right)^n + \left(\frac{\varepsilon}{\eta_S(\varepsilon)}\right)^n \right]. \end{array} \right. $$

<figure><img src="image_117.png" /></figure>
# Idea of the proof

## Global error estimate

$$ \left\{ \begin{array}{l} \| \mathbf{u}^{\varepsilon} - \tilde{\mathbf{u}}_{n}^{\varepsilon} \|_{H^1(\Omega_{\varepsilon}^{R, \delta})} \le C \left[ \left( \eta_H(\varepsilon) \right)^n + \left( \frac{\varepsilon}{\eta_H(\varepsilon)} \right)^n \right] \\ \qquad + C \left[ \left( \eta_S(\varepsilon) \right)^n + \left( \frac{\varepsilon}{\eta_S(\varepsilon)} \right)^n \right]. \end{array} \right. $$

One can choose $\eta_H(\varepsilon)$ and $\eta_S(\varepsilon)$ to **optimize** this relation

$$ \eta_H(\varepsilon) = \eta_S(\varepsilon) = \sqrt{\varepsilon} $$

This leads to

$$ \| \mathbf{u}^{\varepsilon} - \tilde{\mathbf{u}}_{n}^{\varepsilon} \|_{H^1(\Omega_{\varepsilon}^{R, \delta})} \le C \varepsilon^{\frac{n}{2}} $$
Idea of the proof

$$ \| u^{\epsilon} - \tilde{u}_{n}^{\epsilon} \|_{H^{1}(\Omega_{\epsilon}^{R, \delta})} \le C \epsilon^{\frac{n}{2}} $$

<figure><img src="image_118.png" /></figure>
Idea of the proof

$$ \|u^{\epsilon} - \tilde{u}_{n}^{\epsilon}\|_{H^{1}(\Omega_{\epsilon}^{R, \delta})} \le C \epsilon^{\frac{n}{2}} \Rightarrow \|u^{\epsilon} - \tilde{u}_{n}^{\epsilon}\|_{H^{1}(\Omega^{R, R'})} \le C \epsilon^{\frac{n}{2}} $$

<figure><img src="image_119.png" /></figure>
Idea of the proof

$$ \|u^{\epsilon} - \tilde{u}_{n}^{\epsilon}\|_{H^{1}(\Omega_{\epsilon}^{R, \delta})} \le C \epsilon^{\frac{n}{2}} \implies \|u^{\epsilon} - \tilde{u}_{n}^{\epsilon}\|_{H^{1}(\Omega^{R, R'})} \le C \epsilon^{\frac{n}{2}} $$

In the **far field** zone:

$$ \tilde{u}_{n}^{\epsilon} = u_{n}^{H, \epsilon} = u^{0} + \sum_{i=1}^{n} \sum_{k=0}^{i-1} \epsilon^{i} (\log \epsilon)^{k} u_{i}^{k} $$
Idea of the proof

$$ \|u^{\epsilon} - \tilde{u}_{n}^{\epsilon}\|_{H^{1}(\Omega_{\epsilon}^{R, \delta})} \le C \epsilon^{\frac{n}{2}} \implies \|u^{\epsilon} - \tilde{u}_{n}^{\epsilon}\|_{H^{1}(\Omega^{R, R'})} \le C \epsilon^{\frac{n}{2}} $$

In the **far field** zone:

$$ \tilde{u}_n^{\epsilon} = u_{n, \epsilon}^{H, \epsilon} = u^0 + \sum_{i=1}^{n} \sum_{k=0}^{i-1} \epsilon^i (\log \epsilon)^k u_i^k $$

$$ \left\{ \begin{array}{l} \|u^{\epsilon} - u_{3n}^{H, \epsilon}\|_{H^1(\Omega^{R, R'})} \le C \epsilon^{\frac{3n}{2}} \\ \|u_{3n}^{H, \epsilon} - u_{n}^{H, \epsilon}\|_{H^1(\Omega^{R, R'})} \le C \epsilon^{n+1} \log^n \epsilon \end{array} \right. $$
Idea of the proof

$$ \|u^{\varepsilon} - \tilde{u}_{n}^{\varepsilon}\|_{H^{1}(\Omega_{\varepsilon}^{R, \delta})} \le C \varepsilon^{\frac{n}{2}} \implies \|u^{\varepsilon} - \tilde{u}_{n}^{\varepsilon}\|_{H^{1}(\Omega^{R, R'})} \le C \varepsilon^{\frac{n}{2}} $$

In the **far field** zone:

$$ \tilde{u}_{n}^{\varepsilon} = u_{n}^{H, \varepsilon} = u^{0} + \sum_{i=1}^{n} \sum_{k=0}^{i-1} \varepsilon^{i} (\log \varepsilon)^{k} u_{i}^{k} $$

$$ \left\{ \begin{array}{l} \|u^{\varepsilon} - u_{3n}^{H, \varepsilon}\|_{H^{1}(\Omega^{R, R'})} \le C \varepsilon^{\frac{3n}{2}} \\ \|u_{3n}^{H, \varepsilon} - u_{n}^{H, \varepsilon}\|_{H^{1}(\Omega^{R, R'})} \le C \varepsilon^{n+1} \log^{n} \varepsilon \end{array} \right. $$

One can conclude using the **triangular inequality**.
Mathematical analysis

<figure><img src="image_120.png" /></figure>

<figure><img src="image_121.png" /></figure>

<figure><img src="image_122.png" /></figure>

$$ \left\| \textcolor{red}{u^{\epsilon}} - \textcolor{red}{u^0} - \sum_{i=1}^{n} \sum_{k=0}^{i-1} \epsilon^i (\log \epsilon)^k \textcolor{red}{u_i^k} \right\|_{H^1(\Omega^{R,R'})} \le C \epsilon^{n+1} (\log \epsilon)^n \|f\|_{L^2(\Omega)}. $$

$$ \left\| \textcolor{red}{u_p^{\epsilon}} - \sum_{i=0}^{n} \sum_{k=0}^{i} \epsilon^i (\log \epsilon)^k (\textcolor{red}{u_p})_i^k \right\|_{H^1(\mathcal{B}_1^{L,L'})} \le C \epsilon^{n+1} (\log \epsilon)^{n+1} \|f\|_{L^2(\Omega)}. $$

$$ \left\| \textcolor{red}{U^{\epsilon}} - \sum_{i=0}^{n} \sum_{k=0}^{i} \epsilon^i (\log \epsilon)^k \textcolor{red}{U_i^k} \right\|_{H^1(\mathcal{O}_1^{l,l'})} \le C \epsilon^{n+1} (\log \epsilon)^{n+1} \|f\|_{L^2(\Omega)}. $$
1. Mathematical analysis of the finite slot (**resonance** phenomena)

<figure><img src="image_123.png" /></figure>

The difficulty: the **stability** result.
Perspectives

1. Mathematical analysis of the finite slot (**resonance** phenomena)

2. Comparison with the **multi-scale** technique
Perspectives

1. Mathematical analysis of the finite slot (**resonance**) phenomena

2. Comparison with the **multi-scale** technique

3. The 3D **Maxwell** equation

<figure><img src="image_124.png" /></figure>
# Perspectives

1. Mathematical analysis of the finite slot (resonance phenomena)

2. Comparison with the multi-scale technique

3. The 3D Maxwell equation

4. The time domain (evolution equation)

$$ \frac{\partial^2 u}{\partial t^2} - c^2 \Delta u = 0. $$