Abstract
In this paper, we propose and study the uniaxial perfectly matched layer (PML) method for three-dimensional time-domain electromagnetic scattering problems, which has a great advantage over the spherical one in dealing with problems involving anisotropic scatterers. The truncated uniaxial PML problem is proved to be well-posed and stable, based on the Laplace transform technique and the energy method. Moreover, the L2-norm and L∞-norm error estimates in time are given between the solutions of the original scattering problem and the truncated PML problem, leading to the exponential convergence of the time-domain uniaxial PML method in terms of the thickness and absorbing parameters of the PML layer. The proof depends on the error analysis between the EtM operators for the original scattering problem and the truncated PML problem, which is different from our previous work (Wei et al. [SIAM J. Numer. Anal. 58 (2020) 1918–1940]).
Highlights
This paper is concerned with the time-domain electromagnetic scattering by a perfectly conducting obstacle which is modeled by the exterior boundary value problem: ⎧ ⎪ ∇ × E + μ∂tH =0 ⎪ ⎪ ⎪ ⎪ ∇ × H − ε∂tE = J ⎨ n×E =0 E(x, 0) =
For the 3D time-domain electromagnetic scattering problem (1.1a)–(1.1e), the spherical perfectly matched layer (PML) method was proposed in [40] based on the real coordinate stretching technique associated with [Re(s)]−1 in the Laplace transform domain with the Laplace transform variable s ∈ C+, and its exponential convergence was established by means of the energy argument and the exponential decay estimates of the stretched dyadic Green’s function for the Maxwell equations in the free space
We prove the exponential convergence of the uniaxial PML method
Summary
This paper is concerned with the time-domain electromagnetic scattering by a perfectly conducting obstacle which is modeled by the exterior boundary value problem:. A spherical PML method has been proposed in [40] to solve the problem (1.1a)–(1.1e) efficiently, based on the real coordinate stretching technique associated with [Re(s)]−1 in the Laplace transform domain with the Laplace transform variable s ∈ C+ := {s = s1 + is2 ∈ C : s1 > 0, s2 ∈ R}, and its exponential convergence has been established in terms of the thickness and absorbing parameters of the PML layer. For the 3D time-domain electromagnetic scattering problem (1.1a)–(1.1e), the spherical PML method was proposed in [40] based on the real coordinate stretching technique associated with [Re(s)]−1 in the Laplace transform domain with the Laplace transform variable s ∈ C+, and its exponential convergence was established by means of the energy argument and the exponential decay estimates of the stretched dyadic Green’s function for the Maxwell equations in the free space.
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