We present a domain decomposition approach for the computation of the electromagnetic field within periodic structures. We use a Schwarz method with transparent boundary conditions at the interfaces of the domains. Transparent boundary conditions are approximated by the perfectly matched layer method (PML). To cope with Wood anomalies appearing in periodic structures an adaptive strategy to determine optimal PML parameters is developed. We focus on the application to typical EUV lithography line masks. Light propagation within the multi-layer stack of the EUV mask is treated analytically. This results in a drastic reduction of the computational costs and allows for the simulation of next generation lithography masks on a standard personal computer.
Stability and convergence analysis for the domain decomposition finite element/finite difference (FE/FD) method is presented. The analysis is designed for semi-discrete finite element scheme for the time-dependent Maxwell's equations. The explicit finite element schemes in different settings of the spatial domain are constructed and domain decomposition algorithm is formulated. Several numerical examples validate convergence rates obtained in the theoretical studies.
This paper is the first one of two serial articles, whose goal is to prove convergence of HX Preconditioner (proposed by Hiptmair and Xu 2007) for Maxwell's equations with jump coefficients. In this paper we establish various extensions of the regular Helmholtz decomposition for edge finite element functions defined in three dimensional domains. The functions defined by the regular Helmholtz decompositions can preserve the zero tangential complement on faces and edges of polyhedral domains and some non-Lipchitz domains, and possess stability estimates with only a $logarithm$ factor. These regu…
In this work, we consider the time-harmonic Maxwell's equations and their numerical solution with a domain decomposition method. As an innovative feature, we propose a feedforward neural network-enhanced approximation of the interface conditions between the subdomains. The advantage is that the interface condition can be updated without recomputing the Maxwell system at each step. The main part consists of a detailed description of the construction of the neural network for domain decomposition and the training process. To substantiate this proof of concept, we investigate a few subdomains in …
A two-level domain-decomposition preconditioner for the time-harmonic Maxwell's equations
Université Côte d’Azur, CNRS, LJAD（蔚蓝海岸大学，法国国家科学研究中心，拉格朗日分析几何实验室）；University of Strathclyde（思克莱德大学）；University of Bath（巴斯大学）；UPMC Univ Paris 06, LJLL（巴黎第六大学皮埃尔和玛丽·居里大学，拉格朗日分析实验室）
Numerical discretization of the large-scale Maxwell's equations leads to an ill-conditioned linear system that is challenging to solve. The key requirement for successive solutions of this linear system is to choose an efficient solver. In this work we use Perfectly Matched Layers (PML) to increase this efficiency. PML have been widely used to truncate numerical simulations of wave equations due to improving the accuracy of the solution instead of using absorbing boundary conditions (ABCs). Here, we will develop an efficient solver by providing an alternative use of PML as transmission conditi…
The time harmonic Maxwell equations are of current interest in computational science and applied mathematics with many applications in modern physics. In this work, we present parallel finite element solver for the time harmonic Maxwell equations and compare different preconditioners. We show numerically that standard preconditioners like incomplete LU and the additive Schwarz method lead to slow convergence for iterative solvers like generalized minimal residuals, especially for high wave numbers. Even more we show that also more specialized methods like the Schur complement method also yield…
Convergence of parallel overlapping domain decomposition methods with impedance boundary conditions for time-harmonic Maxwell equations in heterogeneous media
The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Shenzhen International Center for Industrial and Applied Mathematics, Shenzhen Research Institute of Big Data（深圳工业与应用数学中心，深圳大数据研究院）；University of Bath（巴斯大学）；Shenzhen Loop Area Institute（深圳河套学院）
Classically transmission conditions between subdomains are optimized for a simplified two subdomain decomposition to obtain optimized Schwarz methods for many subdomains. We investigate here if such a simplified optimization suffices for the magnetotelluric approximation of Maxwell's equation which leads to a complex diffusion problem. We start with a direct analysis for 2 and 3 subdomains, and present asymptotically optimized transmission conditions in each case. We then optimize transmission conditions numerically for 4, 5 and 6 subdomains and observe the same asymptotic behavior of optimize…
An FFT-based algorithm for efficient computation of Green's functions for the Helmholtz and Maxwell's equations in periodic domains
LSEC and Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences（中国科学院数学与系统科学研究院应用数学研究所）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；Center for Industrial Mathematics, University of Bremen（不来梅大学工业数学中心）
In this paper we develop and analyse domain decomposition methods for linear systems of equations arising from conforming finite element discretisations of positive Maxwell-type equations. Convergence of domain decomposition methods rely heavily on the efficiency of the coarse space used in the second level. We design adaptive coarse spaces that complement the near-kernel space made of the gradient of scalar functions. This extends the results in [2] to the variable coefficient case and non-convex domains at the expense of a larger coarse space.
This paper is the second one of two serial articles, whose goal is to prove convergence of HX Preconditioner (proposed by Hiptmair and Xu, 2007) for Maxwell's equations with jump coefficients. In this paper, based on the auxiliary results developed in the first paper (Hu, 2017), we establish a new regular Helmholtz decomposition for edge finite element functions in three dimensions, which is nearly stable with respect to a weight function. By using this Helmholtz decomposition, we give an analysis of the convergence of the HX preconditioner for the case with strongly discontinuous coefficients…
We propose a controllability method for the numerical solution of time-harmonic Maxwell's equations in their first-order formulation. By minimizing a quadratic cost functional, which measures the deviation from periodicity, the controllability method determines iteratively a periodic solution in the time domain. At each conjugate gradient iteration, the gradient of the cost functional is simply computed by running any time-dependent simulation code forward and backward for one period, thus leading to a non-intrusive implementation easily integrated into existing software. Moreover, the propose…
A Bloch decomposition based split-step pseudo spectral method for quantum dynamics with periodic potentials
Tsinghua University（清华大学）；University of Wisconsin, Madison（威斯康星大学麦迪逊分校）；Wolfgang Pauli Institute Vienna（维也纳沃尔夫冈·泡利研究所）；Faculty of Mathematics, University of Vienna（维也纳大学数学学院）
Stochastic domain decomposition for the solution of the two-dimensional magnetotelluric problem
Department of Mathematics and Statistics, Memorial University of Newfoundland（纪念大学数学与统计系）；Department of Earth Sciences, Memorial University of Newfoundland（纪念大学地球科学系）
The Hermite-Taylor method, introduced in 2005 by Goodrich, Hagstrom and Lorenz, is highly efficient and accurate when applied to linear hyperbolic systems on periodic domains. Unfortunately its widespread use has been prevented by the lack of a systematic approach to implementing boundary conditions. In this paper we present the Hermite-Taylor Correction Function method, which provides exactly such a systematic approach for handing boundary conditions. Here we focus on Maxwell's equations but note that the method is easily extended to other hyperbolic problems.
Spectral theory for Maxwell's equations at the interface of a metamaterial. Part I: Generalized Fourier transform
Department of Mathematics of the University of Utah（犹他大学数学系）；ENSTA ParisTech（巴黎高等先进技术学院）；CNRS (Centre National de la Recherche Scientifique)（法国国家科学研究中心）；INRIA (Institut National de Recherche en Informatique et en Automatique)（法国国家计算机科学研究所）
Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption
Sorbonne Université, Université Paris-Diderot SPC, CNRS, Inria, Laboratoire Jacques-Louis Lions（索邦大学，巴黎第七大学SPC，法国国家科学研究中心，法国国家信息与自动化研究所，雅克-路易·利翁斯实验室）；Department of Mathematics and Statistics, University of Strathclyde（斯特拉斯克莱德大学数学与统计系）；Université Côte d’Azur, CNRS, Laboratoire J-A Dieudonné（蔚蓝海岸大学，法国国家科学研究中心，J-A 迪厄多内实验室）；Department of Mathematical Sciences, University of Bath（巴斯大学数学科学系）
A new splitting is proposed for solving the Vlasov-Maxwell system. This splitting is based on a decomposition of the Hamiltonian of the Vlasov-Maxwell system and allows for the construction of arbitrary high order methods by composition (independent of the specific deterministic method used for the discretization of the phase space). Moreover, we show that for a spectral method in space this scheme satisfies Poisson's equation without explicitly solving it. Finally, we present some examples in the context of the time evolution of an electromagnetic plasma instability which emphasizes the excel…
A new domain decomposition method for Maxwell's equations in conductive media is presented. Using this method reconstruction algorithms are developed for determination of dielectric permittivity function using time-dependent scattered data of electric field. All reconstruction algorithms are based on optimization approach to find stationary point of the Lagrangian. Adaptive reconstruction algorithms and space mesh refinement indicators are also presented. Our computational tests show qualitative reconstruction of dielectric permittivity function using anatomically realistic breast phantom.
In this paper we consider a numerical homogenization technique for curl-curl-problems that is based on the framework of the Localized Orthogonal Decomposition and which was proposed in [D. Gallistl, P. Henning, B. Verfürth. SIAM J. Numer. Anal. 56-3:1570-1596, 2018] for problems with essential boundary conditions. The findings of the aforementioned work establish quantitative homogenization results for the time-harmonic Maxwell's equations that hold beyond assumptions of periodicity, however, a practical realization of the approach was left open. In this paper, we transfer the findings from es…
We begin by showing that any $n \times n$ matrix can be decomposed into a sum of $n$ circulant matrices with periodic relaxations on the unit circle. This decomposition is orthogonal with respect to a Frobenius inner product, allowing recursive iterations for these circulant components. It is also shown that the dominance of a few circulant components in the matrix allows sparse similarity transformations using Fast-Fourier-transform (FFT) operations. This enables the evaluation of all eigenvalues of dense Toeplitz, block-Toeplitz, and other periodic or quasi-periodic matrices, to a reasonable…
Domain decomposition for quasi-periodic scattering by layered media via robust boundary-integral equations at all frequencies
MIT（麻省理工学院）；Pontificia Universidad Católica de Chile（智利天主教 Pontificia 大学）；Louisiana State University（路易斯安那州立大学）；New Jersey Inst. of Technology（新泽西理工学院）；Duke University（杜克大学）
This paper is concerned with the inverse scattering problem for the three-dimensional Maxwell's equations in bi-anisotropic periodic structures. The inverse scattering problem aims to determine the shape of bi-anisotropic periodic scatterers from electromagnetic near field data at a fixed frequency. The Factorization method is studied as an analytical and numerical tool for solving the inverse problem. We provide a rigorous justification of the Factorization method which results in the unique determination and a fast imaging algorithm for the periodic scatterer. Numerical examples for imaging …
