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Register a new userConvergence analysis of the PML method for time-domain electromagnetic scattering problems
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Added by
Changkun Wei
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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In this paper, a perfectly matched layer (PML) method is proposed to solve the time-domain electromagnetic scattering problems in 3D effectively. The PML problem is defined in a spherical layer and derived by using the Laplace transform and real coordinate stretching in the frequency domain. The well-posedness and the stability estimate of the PML problem are first proved based on the Laplace transform and the energy method. The exponential convergence of the PML method is then established in terms of the thickness of the layer and the PML absorbing parameter. As far as we know, this is the first convergence result for the time-domain PML method for the three-dimensional Maxwell equations. Our proof is mainly based on the stability estimates of solutions of the truncated PML problem and the exponential decay estimates of the stretched dyadic Greens function for the Maxwell equations in the free space.
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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 $L^2$-norm and $L^{infty}$-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 (SIAM J. Numer. Anal. 58(3) (2020), 1918-1940).
Consider the electromagnetic scattering of a time-harmonic plane wave by an open cavity which is embedded in a perfectly electrically conducting infinite ground plane. This paper is concerned with the numerical solutions of the transverse electric and magnetic polarizations of the open cavity scattering problems. In each polarization, the scattering problem is reduced equivalently into a boundary value problem of the two-dimensional Helmholtz equation in a bounded domain by using the transparent boundary condition (TBC). An a posteriori estimate based adaptive finite element method with the perfectly matched layer (PML) technique is developed to solve the reduced problem. The estimate takes account both of the finite element approximation error and the PML truncation error, where the latter is shown to decay exponentially with respect to the PML medium parameter and the thickness of the PML layer. Numerical experiments are presented and compared with the adaptive finite element TBC method for both polarizations to illustrate the competitive behavior of the proposed method.
In this paper we consider the inverse electromagnetic scattering for a cavity surrounded by an inhomogeneous medium in three dimensions. The measurements are scattered wave fields measured on some surface inside the cavity, where such scattered wave fields are due to sources emitted on the same surface. We first prove that the measurements uniquely determine the shape of the cavity, where we make use of a boundary value problem called the exterior transmission problem. We then complete the inverse scattering problem by designing the linear sampling method to reconstruct the cavity. Numerical examples are further provided to illustrate the viability of our algorithm.
This paper is concerned with the mathematical analysis of the time-domain electromagnetic scattering problem in an infinite rectangular waveguide. A transparent boundary condition is developed to reformulate the problem into an equivalent initial boundary value problem in a bounded domain. The well-posedness and stability are obtained for the reduced problem. The perfectly matched layer method is studied to truncate the waveguide. It is shown that the truncated problem attains a unique solution. Moreover, an explicit error estimate is given between the solutions of the original scattering problem and the truncated problem. Based on the estimate, the stability and exponential convergence are established for the truncated problem. The optimal bound is achieved for the error with explicit dependence on the parameters of the perfectly matched layer.
We introduce an adaptive element-based domain decomposition (DD) method for solving saddle point problems defined as a block two by two matrix. The algorithm does not require any knowledge of the constrained space. We assume that all sub matrices are sparse and that the diagonal blocks are spectrally equivalent to a sum of positive semi definite matrices. The latter assumption enables the design of adaptive coarse space for DD methods that extends the GenEO theory to saddle point problems. Numerical results on three dimensional elasticity problems for steel-rubber structures discretized by a finite element with continuous pressure are shown for up to one billion degrees of freedom.
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