An adaptive finite element DtN method for the open cavity scattering problems

1. Introduction

Consider the electromagnetic scattering of a time-harmonic plane wave by an open cavity, which is referred to as a bounded domain embedded in the ground with its opening aligned with the ground surface. The open cavity scattering problems have significant applications in industry and military. In computational and applied electromagnetics, one of the physical parameter of interests is the radar cross section (RCS), which measures the detectability of a target by a radar system. It is crucial to have a deliberate control in the form of enhancement or reduction of the RCS of a target in the stealth technology. The cavity RCS caused by jet engine inlet ducts or cavity-backed patch or slot antennas can dominate the total RCS of an aircraft or a device. It is indispensable to have a thorough understanding of the electromagnetic scattering characteristic of a target, particularly a cavity, in order to successfully implement any desired control of its RCS.

Due to the important applications, the open cavity scattering problems have received much attention by many researchers in both of the engineering and mathematics communities. The time-harmonic problems of cavity-backed apertures with penetrable material filling the cavity interior were introduced and studied initially by researchers in the engineering community [19, 27, 21]. The mathematical analysis for the well-posedness of the variational problems can be found in [1, 2, 3]

, where the non-local transparent boundary conditions, based on the Fourier transform, were proposed on the open aperture of the cavity. It has been realized that the phenomena of electromagnetic scattering by cavities not only have striking physics but also give rise to many interesting mathematical problems. As more people work on this subject, there has been a rapid development of the mathematical theory and computational methods for the open cavity scattering problems. The stability estimates with explicit dependence on the wavenumber were obtained in

[9, 10]. Various analytical and numerical methods have been proposed to solve the challenging large cavity problem [6, 11, 8, 30, 22]. The overfilled cavity problems, where the filling material inside the cavity may protrude into the space above the ground surface, were investigated in [14, 15, 25, 29], where the transparent boundary conditions, based on the Fourier series, were introduced on a semi-circle enclosing the cavity and filling material. The multiple cavity scattering problem was examined in [24, 33], where the cavity is assumed to be composed of finitely many disjoint components. The mathematical analysis can be found in [7, 12] on the related scattering problems in a locally perturbed half-plane. We refer to the survey [23] and the references cited therein for a comprehensive account on the modeling, analysis, and computation of the open cavity scattering problems.

There are two challenges for the open cavity scattering problems: the problems are formulated in unbounded domains; the solutions may have singularities due to possible nonsmooth surfaces and discontinuous media. In this paper, we present an adaptive finite element method with transparent boundary condition to overcome the difficulties.

The first issue concerns the domain truncation. The unbounded physical domain needs to be truncated into a bounded computational domain. An appropriate boundary condition is required on the artificial boundary of the truncated domain to avoid unwanted wave reflection. Such a boundary condition is known as a transparent boundary condition (TBC). There are two different TBCs for the open cavity scattering problems. For a regular open cavity, where the filling material is inside the cavity, the Fourier transform based TBC is imposed on the open aperture of the cavity; for an overfilled cavity, where the filling material appears to protrude out of the cavity through the open aperture into the space above the ground surface, the Fourier series based TBC is imposed on the semi-circle enclosing the cavity and the protruding part. The latter is adopted in this work since it can be used to handle more general open cavities. We refer to the perfectly matched layer (PML) techniques [33, 34] and the method of boundary integral equations [5] as alternative approaches for dealing with the issue of the unbounded domains of the open cavity scattering problems.

Due to the existence of corners of cavities or the discontinuity of the dielectric coefficient for the filling material, the solutions have singularites that slow down the convergence of the finite element for uniform mesh refinements. The second issue can be resolved by using the a posteriori error estimate based adaptive finite element method. The a posteriori error estimates are computable quantities from numerical solutions. They measure the solution errors of discrete problems without requiring any a priori information of exact solutions. It is known that the meshes and the associated numerical complexity are quasi-optimal for appropriately designed adaptive finite element methods.

The goal of this paper is to combine the adaptive finite element method and the transparent boundary conditions to solve the open cavity scattering problems in an optimal fashion. Specifically, we consider the scattering of a time-harmonic electromagnetic plane wave by an open cavity embedded in an infinite ground plane. Throughout, the medium is assumed to be constant in the $x_{3}$ direction. The ground plane and the cavity wall are assumed to be perfect electric conductors. The cavity is filled with a nonmagnetic and possibly inhomogeneous material, which may protrude out of the cavity to the upper half-space in a finite extend. The infinite upper half-space above the ground plane and the protruding part of the cavity is composed of a homogeneous medium. Two fundamental polarizations, transverse magnetic (TM) and transverse electric (TE), are studied. In this setting, the three-dimensional Maxwell equations may be reduced to the two dimensional Helmholtz equation and generalized Helmholtz equation for TM and TE polarizations, respectively. Based on the Dirichlet-to-Neumann (DtN) map for each polarization, a transparent boundary condition is imposed to reduce the scattering problem equivalently into a boundary value problem in a bounded domain. The nonlocal DtN operator is defined as an infinite Fourier series which needs to be truncated into a sum of finitely many terms in actual computation. The a posteriori error estimate is derived bewteen the solution of original scattering problem and the finite element solution of the discrete problem with the truncated DtN operator. The error estimate takes account of the finite element discretization error and the truncation error of the DtN operator. Using the asymptotic properties of the solution and DtN operator, we consider a dual problem for the error and show that the truncation error of the DtN operator decays exponentially respect to the truncation parameter, which implies that the truncation number does not need to be large. Numerical experiments are presented for both polarization cases to demonstrate the effectiveness of the proposed adaptive method. The related work can be found in [16, 17, 18, 31] on the adaptive finite element DtN method for solving other scattering problems in open domains.

The paper is organized as follows. Section 2 concerns the problem formulation. The three-dimensional Maxwell equations are introduced and reduced into the two-dimensional Helmholtz equation under the two fundamental modes: transverse magnetic (TM) polarization and transverse electric (TE) polarization. Sections 3 and 4 are devoted to the TM and TE polarizations, respectively. In each section, the variational problem and its finite element approximation are introduced; the a posteriori error analysis is given for the discrete problem with the truncated DtN operator; the adaptive finite element algorithm is presented. In Section 5, the stiff matrix is constructed for the the TBC part of the sesquilinear form. Section 6 describes the formulas of the backscatter radar cross section (RCS). Section 7 presents some numerical examples to illustrate the advantages of the proposed method. The paper is concluded with some general remarks and directions for future research in Section 8.

2. Problem formulation

Consider the electromagnetic scattering by an open cavity, which is a bounded domain embedded in the ground with its opening aligned with the ground surface. By assuming the time dependence $e^{- i ω t}$ , the electromagnetic wave propagation is governed by the time-harmonic Maxwell equations

\nabla \times E = i ω B, \nabla \times H = - i ω D + J,

(2.1)

where $E$ is the electric field, $H$ is the magnetic field, $B$ is the magnetic flux density, $D$ is the electric flux density, $J$ is the electric current density, and $ω > 0$ is the angular frequency. For a linear medium, the constitutive relations, describing the macroscopic properties of the medium, are given by

B = μ H, D = ϵ E, J = σ E,

(2.2)

where $μ$ is the magnetic permeability, $ϵ$ is the electric permittivity, and $σ$ is the electrical conductivity. Throughout, the medium is assumed to be non-magnetic, i.e., the magnetic permeability $μ$ is a constant everywhere, but the electric permittivity $ϵ$ and the electrical conductivity $σ$ are allowed to be spatial variable functions. Substituting (2.2) into (2.1) leads to a coupled system for the electric and magnetic fields

\nabla \times E = i ω μ H, \nabla \times H = - i ω ϵ E + σ E

(2.3)

Eliminating the magnetic field from (2.3), we may obtain the Maxwell system for the electric field

\nabla \times (\nabla \times E) - κ^{2} E = 0,

(2.4)

where the wavenumber $κ = (ω^{2} ϵ μ + i ω μ σ)^{1 / 2}, I κ \geq 0$ . Similarly, we may eliminate the electric field and obtain the Maxwell system for the magnetic field

\nabla \times (κ^{- 2} \nabla \times H) - H = 0.

(2.5)

When the cavity has a constant cross section along the $x_{3}$ -axis and the plane of incidence is in the $x_{1} x_{2}$ -plane, as a consequence, the electromagnetic fields are independent of the $x_{3}$ variable. The three-dimensional Maxwell equations can be reduced to either the two-dimensional Helmholtz equation or the two-dimensional generalized Helmholtz equation.

Let $D \subset R^{2}$ be the cross section of the $x_{3}$ -invariant cavity with a Lipschitz continuous boundary $\partial D = S \cup Γ$ . Here $S$ is the cavity wall and $Γ$ is the open aperture of the cavity, which is aligned with the infinite ground plane $Γ_{g}$ . The cavity is filled with an inhomogeneous medium characterized by the dielectric permittivity $ϵ$ , the magnetic permeability $μ$ , and the electric conductivity $σ$ . We point out that the inhomogeneous medium filling the cavity may protrude into the space above the ground plane, which is called an overfilled cavity. Let $B_{R}^{+} = {x \in R^{2} : | x | < R, x_{2} > 0}$ and $B+^R={x∈R2:|x|<^R,x2>0}$ be upper half-discs with radii $R$ and $^R$ , where $R >^R > 0$ . Denote by $Γ_{R}^{+} = {x \in R^{2} : | x | = R, x_{2} > 0}$ and $Γ+^R={x∈R2:|x|=^R,x2>0}$ the upper semi-circles. The radius $^R$ can be taken to be sufficiently large such that the open exterior domain $R_{+}^{2} ∖ B_{^R}^{+}$ is filled with a homogeneous medium with constant permittivity $ϵ = ϵ_{0}$ and zero conductivity $σ = 0$ . Let $Ω = B_{R}^{+} \cup D$ be the bounded domain where our reduced boundary value problems are formulated. The problem geometry is shown in Figure 1.

Figure 1. Problem geometry of the electromagnetic scattering by an open cavity.

Since the structure is invariant in the $x_{3}$ -axis, we consider two fundamental polarizations for the electromagnetic fields: transverse magnetic (TM) polarization and transverse electric (TE) polarization. In TM case, the magnetic field is perpendicular to the plane of incidence and does not have the component in the $x_{3}$ -axis; the electric field, being perpendicular to the magnetic field and lying in the $x_{1} x_{2}$ -plane, is invariant in the $x_{3}$ -axis and takes the form $E (x_{1}, x_{2}) = (0, 0, u (x_{1}, x_{2}))$ , where $u$ is a scalar function. It is easy to verify from (2.4) that $u$ satisfies the Helmholtz equation

Δ u + κ^{2} u = 0 in R_{+}^{2} \cup D .

(2.6)

In TE case, the electromagnetic fields are characterized by its electric field being perpendicular to the plane of incidence and contain no electric field component in the $x_{3}$ -axis. The magnetic field, being perpendicular to the electric field and lying in the $x_{1} x_{2}$ -plane, is invariant in the $x_{3}$ -axis and has the form $H (x_{1}, x_{2}) = (0, 0, u (x_{1}, x_{2}))$ , where $u$ is also a scalar function. It follows from (2.5) that $u$ satisfies the generalized Helmholtz equation

\nabla \cdot (κ^{- 2} \nabla u) + u = 0 in R_{+}^{2} \cup D .

(2.7)

When the ground plane and the cavity wall are assumed to be perfect conductors, the following perfectly electrically conducting (PEC) boundary condition can be imposed

ν \times E = 0 on Γ_{g} \cup S,

(2.8)

where $ν$

is the unit normal vector to

Γ_{g}

and

S

. In TM polarization, the PEC boundary condition (2.8) reduces to the homogeneous Dirichlet boundary condition

u = 0 on Γ_{g} \cup S .

(2.9)

In TE polarization, the PEC boundary condition (2.8) reduces to the homogeneous Neumann boundary condition

\partial_{ν} u = 0 on Γ_{g} \cup S .

(2.10)

In this paper, we consider the numerical solutions and present an adaptive finite element DtN method for both of the TM problem (2.6), (2.9) and the TE problem (2.7), (2.10). The more complicated three-dimensional Maxwell equations will be our future work.

3. TM polarization

In this section, we discuss the TM polarization and study its finite element approximation. The a posteriori analysis is carried out for both the finite element discretization error and the DtN operator truncation error. An adaptive finite element DtN method is presented for the truncated discrete problem.

3.1. Variational problem

In TM polarization, the nonzero component of the electric field $u$ satisfies the boundary value problem

{\begin{matrix} Δ u + κ^{2} u = 0 & i n R_{+}^{2} \cup D, u = 0 & o n Γ_{g} \cup S . \end{matrix}

(3.1)

Since the problem is imposed in the open domain, a radiation condition is required to complete the formulation.

Consider the incidence of a plane wave

u^{i} (x_{1}, x_{2}) = e^{i (α x_{1} - β x_{2})},

which is sent from the above to impinge the cavity. Here $α = κ_{0} sin θ, β = κ_{0} cos θ$ , $θ \in (- \frac{π}{2}, \frac{π}{2})$ is the incident angle, and $κ_{0} = ω (ϵ_{0} μ)^{1 / 2}$ is the wavenumber in the free space $R_{+}^{2} ∖ B_{R}^{+}$ . It is easy to verify from (2.9) that the reflected wave is

u^{r} (x_{1}, x_{2}) = - e^{i (α x_{1} + β x_{2})} .

By the Jacobi–Anger identity, the incident and reflected waves admit the following expansions:

u^{i} (x) = J_{0} (κ_{0} r) + 2 \infty \sum n = 1 i^{n} J_{n} (κ_{0} r) cos n (θ - π / 2 - ϕ)

(3.2)

and

u^{r} (x) = - J_{0} (κ_{0} r) - 2 \infty \sum n = 1 i^{n} J_{n} (κ_{0} r) cos n (θ - π / 2 + ϕ),

(3.3)

where $J_{n}$ is the Bessel function of the first kind with order $n$ and $x = r (cos ϕ, sin ϕ)$ with $ϕ$ being the observation angle. Define the reference wave $u^{r e f} = u^{i} + u^{r}$ . It follows from (3.2)–(3.3) that

u^{r e f} (x) = \infty \sum n = 1 4 i^{n} J_{n} (κ_{0} r) sin n (θ - π / 2) sin n ϕ .

(3.4)

The total field $u$ consists of the reference field $u^{r e f}$ and the scattered field $u^{s}$ , i.e.,

u = u^{r e f} + u^{s},

(3.5)

where the scattered field $u^{s}$ is required to satisfy the Sommerfeld radiation condition

lim r = | x | \to \infty r^{1 / 2} (\partial_{r} u^{s} - i κ_{0} u^{s}) = 0.

Let $L_{T M}^{2} (Γ_{R}^{+}) := {u \in L^{2} (Γ_{R}^{+}) : u (R, 0) = u (R, π) = 0}$ . For any $u \in L_{T M}^{2} (Γ_{R}^{+})$ , it has the Fourier series expansion

u (R, ϕ) = \infty \sum n = 1 a_{n} sin n ϕ, a_{n} = \frac{2}{π} \int_{0}^{π} u (R, ϕ) sin n ϕ d ϕ .

Define the trace function space $H_{T M}^{s} (Γ_{R}^{+}) := {u \in L_{T M}^{2} (Γ_{R}^{+}) : ∥ u ∥_{H_{T M}^{s} (Γ_{R}^{+})} \leq \infty},$ where the $H_{T M}^{s} (Γ_{R}^{+})$ norm is given by

∥ u ∥_{H_{T M}^{s} (Γ_{R}^{+})} = {(\infty \sum n = 1 (1 + n^{2})^{s} | a_{n} |^{2})}_{n}^{1 / 2} .

It is clear that the dual space of $H_{T M}^{s} (Γ_{R}^{+})$ is $H_{T M}^{- s} (Γ_{R}^{+})$ with respect to the scalar product in $L^{2} (Γ_{R}^{+})$ given by

⟨ u, v ⟩_{Γ_{R}^{+}} = \int_{Γ_{R}^{+}} u ¯ ¯ ¯ v d s .

As discussed in [23], a DtN operator is introduced on $Γ_{R}^{+}$ :

(B_{T M} u) (R, ϕ) = κ_{0} \infty \sum n = 1 \frac{H_{n}^{(1)^{'}} (κ_{0} R)}{H_{n}^{(1)} (κ_{0} R)} a_{n} sin n ϕ,

(3.6)

where $H_{n}^{(1)}$ is the Hankel function of the first kind with order $n$ . It is shown in [29, Lemma 3.1] that $B_{T M} : H_{T M}^{1 / 2} (Γ_{R}^{+}) \to H_{T M}^{- 1 / 2} (Γ_{R}^{+})$ is continuous. The TBC can be imposed for the total field as follows:

\partial_{ρ} u = B_{T M} u + f o n Γ_{R}^{+},

where $f = \partial_{ρ} u^{r e f} - B_{T M} u^{r e f}$ . Substituting (3.4) into (3.6) and applying the Wronskian identity, we obtain explicitly

f = - \frac{8}{π R} \infty \sum n = 1 \frac{i^{n + 1}}{} H_{n}^{(1)} (κ_{0} R) sin n (θ - π / 2) sin n ϕ .

The original cavity scattering problem (3.1) can be reduced equivalently into the boundary value problem

⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} Δ u + k^{2} u = 0 & i n Ω, u = 0 & o n S \cup Γ_{g}, \partial_{ρ} u - B_{T M} u = f & o n Γ_{R}^{+}, \end{matrix}

which has the variational formulation: find $u \in H_{0}^{1} (Ω) := {u \in H^{1} (Ω) : u = 0 o n S \cup Γ_{g}}$ such that

a_{T M} (u, v) = \int_{Γ_{R}^{+}} f ¯ v d s \forall v \in H_{0}^{1} (Ω) .

(3.7)

Here the sesquilinear form $a_{T M} : H_{0}^{1} (Ω) \times H_{0}^{1} (Ω) \to C$ is defined as

a_{T M} (u, v) = \int_{Ω} \nabla u \cdot \nabla ¯ v d x - \int_{Ω} κ^{2} u ¯ v d x - \int_{Γ_{R}^{+}} B_{T M} u ¯ v d s .

The following theorem of the well-posedness of the variational problem (3.7) is proved in [23].

Theorem 3.1.

The variational problem (3.7) has a unique solution $u \in H_{0}^{1} (Ω)$ , which satisfies the estimate

∥ u ∥_{H^{1} (Ω)} ≲ ∥ f ∥_{H^{- 1 / 2} (Γ_{R}^{+})} .

Hereafter, the notation $a ≲ b$ stands for $a \leq C b$ , where $C$ is a positive constant whose value is not required but should be clear from the context.

3.2. Finite element approximation

Let $M_{h}$ be a regular triangulation of $Ω$ , where $h$ denotes the maximum diameter of all the elements in $M_{h}$ . To avoid being distracted from the main focus of the a posteriori error analysis, we assume for simplicity that $S$ and $Γ_{R}^{+}$ are polygonal to keep from using the isoparametric finite element space and deriving the approximation error of the boundaries $S$ and $Γ_{R}^{+}$ . Thus any edge $e \in M_{h}$ is a subset of $\partial Ω$ if it has two boundary vertices.

Let $V_{h} \subset H_{0}^{1} (Ω)$ be a conforming finite element space, i.e.,

V_{h} := {v_{h} \in C (¯ ¯¯ ¯ Ω) : v_{h} |_{T} \in P_{m} (K) \forall T \in M_{h}, v_{h} = 0 o n S \cup Γ_{g}} .

In practice, the DtN operator (3.6) needs to be truncated into a sum of finitely many terms

B_{T M}^{N} u = κ_{0} N \sum n = 1 \frac{H_{n}^{(1)^{'}} (κ_{0} R)}{H_{n}^{(1)} (κ_{0} R)} a_{n} sin n ϕ, a_{n} = \frac{2}{π} \int_{0}^{π} u (R, ϕ) sin n ϕ d ϕ .

(3.8)

Taking account of the DtN operator truncation, we obtain the finite element approximation to the variational problem (3.7): find $u_{h} \in V_{h}$ such that

a_{T M}^{N} (u^{h}, v^{h}) = \int_{Γ_{R}^{+}} f^{h} d s \forall v_{h} \in V_{h},

(3.9)

where the sesquilinear form $a_{T M}^{N} : V_{h} \times V_{h} \to C$ is

a_{T M}^{N} (u^{h}, v^{h}) = \int_{Ω} \nabla u^{h} \cdot \nabla^{h} d x - \int_{Ω} κ^{2} u^{h}^{h} d x - \int_{Γ_{R}^{+}} B_{T M}^{N} u^{h}^{h} d s .

For sufficiently small $h$ and sufficiently large $N$ , the discrete inf-sup condition of the sesquilinear form $a_{T M}^{N}$ can be established by an argument of Schatz [28]. It follows from the general theory in [4] that the truncated variational problem (3.9) admits a unique solution. Since our focus is the a posteriori error estimate and the associated adaptive algorithm, we assume that the discrete problem (3.9) has a unique solution $u_{h} \in V_{h}$ .

3.3. A posteriori error analysis

For any triangular element $T \in M_{h}$ , denoted by $h_{T}$ its diameter. Let $B_{h}$ denote the set of all the edges of $T$ . For any edge $e \in B_{h}$ , denote by $h_{e}$ its length. For any interior edge $e$ , which is the common side of triangular elements $T_{1}, T_{2} \in M_{h}$ , we define the jump residual across $e$ as

J_{e} = - (\nabla u^{h} |_{T_{1}} \cdot ν_{1} + \nabla u^{h} |_{T_{2}} \cdot ν_{2}),

where $ν_{j}$ is the unit outward normal vector on the boundary of $T_{j}, j = 1, 2$ . For any boundary edge $e \subset Γ_{R}^{+}$ , the jump residual is defined as

J_{e} = 2 (B_{T M}^{N} u^{h} - \nabla u^{h} \cdot ν - f) .

For any triangle $T \in M_{h}$ , denote by $η_{T}$ the local error estimator as follows:

η_{T} = h_{T} ∥ H_{T M} u^{h} ∥_{L^{2} (T)} + (\frac{1}{2} \sum e \in \partial T h_{e} ∥ J_{e} ∥_{L^{2} (e)}^{2})^{1 / 2}

where $H_{T M}$ is the Helmholtz operator defined by $H_{T M} u = Δ u + κ^{2} u$ .

Let $ξ = u - u^{h}$ , where $u$ and $u_{h}$ are the solutions of the variational problems (3.7) and (3.9), respectively. Introduce a dual problem: find $w \in H_{0}^{1} (Ω)$ such that

a_{T M} (v, w) = \int_{Ω} v ¯ ξ d x \forall v \in H_{0}^{1} (Ω) .

(3.10)

It is easy to check that $w$ is the solution of the following boundary value problem:

⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} Δ w + κ^{2} w = - ξ & i n Ω, w = 0 & o n S \cup Γ_{g}, \partial_{ρ} w = B_{T M}^{*} w & o n Γ_{R}^{+}, \end{matrix}

where $B_{T M}^{*}$ is the adjoint operator of $B_{T M}$ and is given by

B_{T M}^{*} u = κ_{0} \infty \sum n = 1 ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ ⎛ ⎝ \frac{H_{n}^{(1)^{'}} (κ_{0} R)}{H_{n}^{(1)} (κ_{0} R)} ⎞ ⎠ a_{n} sin n ϕ, a_{n} = \frac{2}{π} \int_{0}^{π} u (R, ϕ) sin n ϕ d ϕ .

The following three lemmas are proved in [17], where the first lemma concerns the well-posedness of the dual problem, the second lemma gives the trace result in $H_{0}^{1} (Ω)$ , and the third lemma shows the error representation formulas.

Lemma 3.2.

The dual problem (3.10) has a unique solution $w \in H_{0}^{1} (Ω)$ , which satisfies the estimate

∥ w ∥_{H^{1} (Ω)} ≲ ∥ ξ ∥_{L^{2} (Ω)} .

Lemma 3.3.

For any $u \in H_{0}^{1} (Ω)$ , the following estimates hold:

∥ u ∥_{H^{1 / 2} (Γ_{R}^{+})} ≲ ∥ u ∥_{H^{1} (Ω)}, ∥ u ∥_{H^{1 / 2} (Γ_{^R}^{+})} ≲ ∥ u ∥_{H^{1} (Ω)} .

Lemma 3.4.

Let $u, u_{h}$ , and $w$ be the solutions to the problems (3.7), (3.9), and (3.10), respectively. The following identities hold:

	$∥ ξ ∥_{H^{1} (Ω)}^{2} = R (a_{T M} (ξ, ξ) + ⟨ (B_{T M}^{N} - B_{T M}) ξ, ξ ⟩_{Γ_{R}^{+}}) + R ⟨ B_{T M}^{N} ξ, ξ ⟩_{Γ_{R}^{+}} + R \int_{Ω} (κ^{2} + 1) \| ξ \|^{2} d x,$
	$∥ ξ ∥_{L^{2} (Ω)}^{2} = a_{T M} (ξ, w) + ⟨ (B_{T M} - B_{T M}^{N}) ξ, w ⟩_{Γ_{R}^{+}} - ⟨ (B_{T M} - B_{T M}^{N}) ξ, w ⟩_{Γ_{R}^{+}},$
	$a_{T M} (ξ, ψ) + ⟨ (B_{T M} - B_{T M}^{N}) ξ, ψ ⟩_{Γ_{R}^{+}} = \int_{Γ_{R}^{+}} f (¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ ψ - ψ_{h}) d s - a_{T M}^{N} (u^{h}, ψ - ψ^{h})$
	$+ ⟨ (B_{T M} - B_{T M}^{N}) u, ψ ⟩_{Γ_{R}^{+}} \forall ψ \in H_{0}^{1} (Ω), ψ_{h} \in V_{h} .$

The following result concerns the truncation error of the DtN operator and plays an important role in the a posteriori error estimate.

Lemma 3.5.

Let $u$ be the solution to (3.7) and $ψ$ be any function in $H_{0}^{1} (Ω)$ . For sufficiently large $N$ , the following estimate holds:

∣ ∣ ⟨ (B_{T M} - B_{T M}^{N}) u, ψ ⟩_{Γ_{R}^{+}} ∣ ∣ ≲ [(\frac{^R}{R})^{N} + (\frac{e κ_{0} R}{} 2 N)^{2 N + 4}] ∥ u^{r e f} ∥_{H^{1} (Ω)} ∥ ψ ∥_{H^{1} (Ω)} .

Proof.

By (3.5), we have $u = u^{s} + u^{r e f}$ , where $u^{r e f}$ is the reference field and $u^{s}$ is the scattered field satisfying the Sommerfeld radiation condition. For sufficiently large $N$ , it is shown in [17, Lemma 4] that

| ⟨ (B_{T M} - B_{T M}^{N}) u^{s}, ψ ⟩_{Γ_{R}^{+}} | ≲ (\frac{^R}{R})^{N} ∥ u^{s} ∥_{H^{1 / 2} (Γ_{R}^{+})} ∥ ψ ∥_{H^{1 / 2} (Γ_{R}^{+})} .

A straightforward calculation yields

(B_{T M} - B_{T M}^{N}) u^{r e f} = κ_{0} \infty \sum n = N + 1 \frac{H_{n}^{(1)^{'}} (κ_{0} R)}{H_{n}^{(1)} (κ_{0} R)} [4 i^{n} J_{n} (κ_{0} R) sin n (θ - \frac{π}{2})] sin n ϕ .

By [32], for sufficiently large $n$ , we have

J_{n} (z) \sim \frac{1}{\sqrt{2 π n}} {(\frac{e z}{2 n})}^{n}, ∣ ∣ ∣ ∣ \frac{H_{n}^{(1)^{'}} (κ_{0} R)}{H_{n}^{(1)} (κ_{0} R)} ∣ ∣ ∣ ∣ ≲ n,

which give

	$\| ⟨ (B_{T M} - B_{T M}^{N}) u^{r e f}, ψ ⟩ \| = \frac{}{π} 2 κ_{0} ∣ ∣ ∣ ∣ \infty \sum n = N + 1 \frac{H_{n}^{(1)^{'}} (κ_{0} R)}{H_{n}^{(1)} (κ_{0} R)} [4 i^{n} J_{n} (κ_{0} R) sin n (θ - \frac{π}{2})] {^ψ}_{n} (R) ∣ ∣ ∣ ∣$
	$\leq 2 π κ_{0} \infty \sum n = N + 1 ∣ ∣ ∣ ∣ \frac{H_{n}^{(1)^{'}} (κ_{0} R)}{H_{n}^{(1)} (κ_{0} R)} ∣ ∣ ∣ ∣ \| J_{n} (κ_{0} R) \| \| {^ψ}_{n} (R) \|$
	$≲ 2 π κ_{0} \infty \sum n = N + 1 n \frac{1}{\sqrt{2 π n}} {(\frac{e κ_{0} R}{2 n})}^{n} \| {^ψ}_{n} (R) \|$
	$≲√2πκ0⎧⎨⎩∞∑n=N+1[√n1√2πn(eκ0R2n)n]2⎫⎬⎭1/2{∞∑n=N+12π(1+n2)1/2\|^ψn(R)\|2}1/2$
	$≲ κ_{0} {\infty \sum n = N + 1 {(\frac{e κ_{0} R}{2 n})}^{2 n}}^{1 / 2} ∥ ψ ∥_{H^{1 / 2} (Γ_{R}^{+})} .$

For $N > \frac{e κ_{0} R}{2}$ , it is easy to verify

\infty \sum n = N + 1 {(\frac{e κ_{0} R}{2 n})}^{2 n} \leq \infty \sum n = N + 1 {(\frac{e κ_{0} R}{2 N})}^{2 n} = \frac{{(\frac{e κ_{0} R}{2 N})}^{2 N + 4}}{1 - {(\frac{e κ_{0} R}{2 N})}^{2}} .

Hence

$\| ⟨ (B_{T M} - B_{T M}^{N}) u^{r e f}, ψ ⟩ \|$	$≲$	$κ_{0} \frac{{(\frac{e κ_{0} R}{2 N})}^{2 N + 4}}{1 - {(\frac{e κ_{0} R}{2 N})}^{2}} \sqrt{2 π R^{2}} \sqrt{\frac{1}{2 π R^{2}}} ∥ ψ ∥_{H^{1 / 2} (Γ_{R}^{+})}$
	$≲$	$\frac{κ_{0}}{\sqrt{2 π} R} \frac{{(\frac{e κ_{0} R}{2 N})}^{2 N + 4}}{1 - {(\frac{e κ_{0} R}{2 N})}^{2}} ∥ u^{r e f} ∥_{H^{1} (B_{R}^{+})} ∥ ψ ∥_{H^{1} (Ω)}$
	$\leq$	$\frac{κ_{0}}{\sqrt{2 π} R} \frac{1}{1 - {(\frac{e κ_{0} R}{2 N})}^{2}} {(\frac{e κ_{0} R}{2 N})}^{2 N + 4} ∥ u^{r e f} ∥_{H^{1} (Ω)} ∥ ψ ∥_{H^{1} (Ω)} .$

Combining the above estimates, we obtain

| ⟨ (B_{T M} - B_{T M}^{N}) u, ψ ⟩ | ≲ {(\frac{R^{'}}{R})}^{N} ∥ u^{s} ∥_{H^{1 / 2} (Γ_{R}^{+})} ∥ ψ ∥_{H^{1} (Ω)} + {(\frac{e κ_{0} R}{2 N})}^{2 N + 4} ∥ u^{r e f} ∥_{H^{1} (Ω)} ∥ ψ ∥_{H^{1} (Ω)} .

Since

∥ u^{s} ∥_{H^{1} (Ω)} = ∥ u - u^{r e f} ∥_{H^{1} (Ω)} \leq ∥ u ∥_{H^{1} (Ω)} + ∥ u^{r e f} ∥_{H^{1} (Ω)}

and

∥ u ∥_{H^{1} (Ω)} ≲ ∥ f ∥_{H^{- 1 / 2} (Γ_{R}^{+})},

it suffices to estimate $f$ . A simple calculation yields

	$∥ f ∥_{H^{- 1 / 2} (Γ_{R}^{+})}$	$\leq$	$∥ \partial_{ρ} u^{r e f} ∥_{H^{- 1 / 2} (Γ_{R}^{+})} + ∥ B_{T M} u^{r e f} ∥_{H^{- 1 / 2} (Γ_{R}^{+})}$
		$≲$	$∥ \partial_{ρ} u^{r e f} ∥_{H^{- 1 / 2} (Γ_{R}^{+})} + ∥ u^{r e f} ∥_{H^{1 / 2} (Γ_{R}^{+})} .$

Taking the normal derivative of (3.4), we get

\partial_{ρ} u^{r e f} = 4 \infty \sum n = 1 i^{n} κ_{0} J_{n}^{'} (κ_{0} R) sin n (θ - \frac{π}{2}) sin n ϕ .

It follows from the definition of the norm on $H^{- 1 / 2} (Γ_{R}^{+})$ that

For sufficiently large $n$ , we have from the asymptotic property of the Bessel function $J_{n}$ (cf. [32]) that

\frac{J_{n}^{'} (z)}{J_{n} (z)} = \frac{J_{n - 1} (z)}{J_{n} (z)} - \frac{n}{z} \sim \frac{\sqrt{n}}{\sqrt{n - 1}} {(\frac{n}{n - 1})}^{n - 1} \frac{2 n}{e z} - \frac{}{n} z \sim \frac{n}{z} .

Hence

$∥ \partial_{ρ} u^{r e f} ∥_{H^{- 1 / 2} (Γ_{R}^{+})}$	$≲$	$2 π {\infty \sum n = 1 {(1 + n^{2})}^{- 1 / 2} κ_{0}^{2} n^{2} {∣ ∣ ∣ J_{n} (κ_{0} R) sin n (θ - \frac{π}{2}) ∣ ∣ ∣}_{n}^{2}}^{1 / 2}$
	$≲$	${2 π \infty \sum n = 1 {(1 + n^{2})}^{1 / 2} κ_{0}^{2} {\| J_{n} (κ_{0} R) \|}_{n}^{2} {∣ ∣ ∣ sin n (θ - \frac{π}{2}) ∣ ∣ ∣}^{2}}^{1 / 2}$
	$=$	$∥ u^{r e f} ∥_{H^{1 / 2} (Γ_{R}^{+})} .$

Noting

∥ f ∥_{H^{- 1 / 2} (Γ_{R}^{+})} ≲ ∥ u^{r e f} ∥_{H^{1 / 2} (Γ_{R}^{+})} ≲ ∥ u^{r e f} ∥_{H^{1} (Ω)},

we complete the proof. ∎

Remark 3.6.

We notice that the result and proof of Lemma 3.5 is different from those for the scattering problems in periodic structures [18, 31, 26]. For the latter problems, the DtN operators are defined on a straight line or plane surface and have only finitely many terms when acting on the incident fields. For our case, the DtN operator is defined on a semi-circle and is still an infinite series when acting on the reference field, which results in an extra term in the estimate given in Lemma 3.5.

Lemma 3.7.

Let $w$ be the solution of the dual problem (3.10). Then the following estimate holds:

| ⟨ (B_{T M} - B_{T M}^{N}) ξ, w ⟩_{Γ_{R}^{+}} | \leq N^{- 2} ∥ ξ ∥_{H^{1} (Ω)}^{2} .

Proof.

Since $w = 0$ on $Γ_{g}$ , it admits the Fourier series expansion in terms of the sin functions

w (r, ϕ) = \infty \sum n = 1 {^w}^{(n)} (r) sin n ϕ, r \in [R^{'}, R], ϕ \in [0, π],

where $w^{(n)} (r)$ are the Fourier coefficients. Following the same proof as that in [17, Lemma 5], we may show the desired result. ∎

The following theorem presents the a posteriori error estimate and is the main result for the TM polarization. By Lemma (3.5), the proof is essentially the same as that for [17, Theorem 1]. The details are omitted for brevity.

Theorem 3.8.

Let $u$ and $u_{h}$ be the solution of (3.7) and (3.9), respectively. There exists a positive integer $N_{0}$ independent of $h$ such that for $N > N_{0}$ , the following a posteriori error estimate holds:

∥ u - u^{h} ∥_{H^{1} (Ω)} ≲ (\sum T \in M_{h} η_{T}^{2})^{1 / 2} + [(\frac{^R}{R})^{N} + (\frac{e κ_{0} R}{2 N})^{2 N + 4}] ∥ u^{r e f} ∥_{H^{1} (Ω)} .

3.4. Adaptive FEM algorithm

It is shown in Theorem 3.8 that the a posteriori error consists of two parts: the finite element discretization error $ε_{h}$ and the DtN operator truncation error $ε_{N}$ , where

(3.11)

In the implementation, based on (3.11), the parameters $^R, R$ , and $N$ can be chosen appropriately such that the finite element discretization error is not contaminated by the truncation error, i.e., $ε_{N}$ is required to be small compared with $ε_{h}$ , for instance, $ε_{N} \leq 10^{- 8}$ . Table 1 shows the algorithm of the adaptive finite element DtN method for solving the open cavity scattering problem in the TM polarization.

Given the tolerance $ε > 0$ and the parameter $τ \in (0, 1)$ .
Fix the computational domain $Ω$ by choosing $R$ .
Choose $R^{'}$ and $N$ such that $ϵ_{N} \leq 10^{- 8}$ .
Construct an initial triangulation $M_{h}$ over $Ω$ and compute error estimators.
While $ε_{h} > ε$ do
refine mesh $M_{h}$ according to the strategy

$if η_{^T} > τ max T \in M_{h} η_{T}, % refine the element^T \in M_{h},$
denote refined mesh still by $M_{h}$ , solve the discrete problem (3.9) on the new mesh $M_{h}$ ,
compute the corresponding error estimators.
End while.

Table 1. The adaptive finite element DtN method for TM polarization.

4. TE polarization

In this section, we consider the TE polarization. Since the discussions are similar to the TM polarization, we briefly present the parallel results without providing the details.

In TE polarization, the total field $u$ satisfies the boundary value problem of the generalized Helmholtz equation

{\begin{matrix} \nabla \cdot (κ^{- 2} \nabla u) + u = 0 & i n Ω, \partial_{ν} u = 0 & o n S \cup Γ_{g} . \end{matrix}

(4.1)

Consider the same plane incident wave $u^{i} (x_{1}, x_{2}) = e^{i (α x_{1} - β x_{2})}$ . Due to the homogeneous Neumann boundary condition on $Γ_{g}$ , the reflected field is

u^{r} (x_{1}, x_{2}) = e^{i (α x_{1} + β x_{2})} .

By the Jacobi–Anger identity, the reference wave $u^{r e f} = u^{i} + u^{r}$ admits the following expansion:

u^{r e f} (x_{1}, x_{2}) = 2 J_{0} (κ_{0} r) + 4 \infty \sum n = 1 i^{n} J_{n} (κ_{0} r) cos n (θ - π / 2) cos n ϕ .

(4.2)

Once again, the total field $u$ is assumed to be composed of the reference field

	$\| ⟨ (B_{T M} - B_{T M}^{N}) u^{r e f}, ψ ⟩ \| = \frac{}{π} 2 κ_{0} ∣ ∣ ∣ ∣ \infty \sum n = N + 1 \frac{H_{n}^{(1)^{'}} (κ_{0} R)}{H_{n}^{(1)} (κ_{0} R)} [4 i^{n} J_{n} (κ_{0} R) sin n (θ - \frac{π}{2})] {^ψ}_{n} (R) ∣ ∣ ∣ ∣$
	$\leq 2 π κ_{0} \infty \sum n = N + 1 ∣ ∣ ∣ ∣ \frac{H_{n}^{(1)^{'}} (κ_{0} R)}{H_{n}^{(1)} (κ_{0} R)} ∣ ∣ ∣ ∣ \| J_{n} (κ_{0} R) \| \| {^ψ}_{n} (R) \|$
	$≲ 2 π κ_{0} \infty \sum n = N + 1 n \frac{1}{\sqrt{2 π n}} {(\frac{e κ_{0} R}{2 n})}^{n} \| {^ψ}_{n} (R) \|$
	$≲√2πκ0⎧⎨⎩∞∑n=N+1[√n1√2πn(eκ0R2n)n]2⎫⎬⎭1/2{∞∑n=N+12π(1+n2)1/2\|^ψn(R)\|2}1/2$
	$≲ κ_{0} {\infty \sum n = N + 1 {(\frac{e κ_{0} R}{2 n})}^{2 n}}^{1 / 2} ∥ ψ ∥_{H^{1 / 2} (Γ_{R}^{+})} .$

An adaptive finite element DtN method for the open cavity scattering problems

Authors

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Spectral Galerkin method for solving elastic wave scattering problems with multiple open arcs

1. Introduction

2. Problem formulation

3. TM polarization

3.1. Variational problem

Theorem 3.1.

3.2. Finite element approximation

3.3. A posteriori error analysis

Lemma 3.2.

Lemma 3.3.

Lemma 3.4.

Lemma 3.5.

Proof.

Remark 3.6.

Lemma 3.7.

Proof.

Theorem 3.8.

3.4. Adaptive FEM algorithm

4. TE polarization

An adaptive finite element DtN method for the open cavity scattering problems

Authors

Related Research

An adaptive finite element PML method for the open cavity scattering problems

An adaptive finite element DtN method for the elastic wave scattering by biperiodic structures

An adaptive finite element DtN method for the three-dimensional acoustic scattering problem

An adaptive finite element DtN method for the elastic wave scattering problem in three dimensions

A posteriori local error estimation for finite element solutions of boundary value problems

On the half-space matching method for real wavenumber

Spectral Galerkin method for solving elastic wave scattering problems with multiple open arcs

This week in AI

1. Introduction

2. Problem formulation

3. TM polarization

3.1. Variational problem

Theorem 3.1.

3.2. Finite element approximation

3.3. A posteriori error analysis

Lemma 3.2.

Lemma 3.3.

Lemma 3.4.

Lemma 3.5.

Proof.

Remark 3.6.

Lemma 3.7.

Proof.

Theorem 3.8.

3.4. Adaptive FEM algorithm

4. TE polarization