We propose and analyze a discontinuous least squares finite element method for solving the indefinite time-harmonic Maxwell equations. The scheme is based on the $L^2$ norm least squares functional with the weak imposition of the continuity across the interior faces. We minimize the functional over the piecewise polynomial spaces to seek numerical solutions. The method is shown to be stable without any constraint on the mesh size. We prove the convergence orders under both the energy norm and the $L^2$ norm. Numerical results in two and three dimensions are presented to verify the error estimates.
In this paper we propose and analyze finite element discontinuous Galerkin methods for the one- and two-dimensional stochastic Maxwell equations with multiplicative noise. The discrete energy law of the semi-discrete DG methods were studied. Optimal error estimate of the semi-discrete method is obtained for the one-dimensional case, and the two-dimensional case on both rectangular meshes and triangular meshes under certain mesh assumptions. Strong Taylor 2.0 scheme is used as the temporal discretization. Both one- and two-dimensional numerical results are presented to validate the theoretical …
In this paper we are concerned with plane wave discontinuous Galerkin (PWDG) methods for Helmholtz equation and time-harmonic Maxwell equations in three-dimensional anisotropic media, for which the coefficients of the equations are matrices instead of numbers. We first define novel plane wave basis functions based on rigorous choices of scaling transformations and coordinate transformations. Then we derive the error estimates of the resulting approximate solutions with respect to the condition number of the coefficient matrices, under a new assumption on the shape regularity of polyhedral mesh…
One- and multi-dimensional stochastic Maxwell equations with additive noise are considered in this paper. It is known that such system can be written in the multi-symplectic structure, and the stochastic energy increases linearly in time. High order discontinuous Galerkin methods are designed for the stochastic Maxwell equations with additive noise, and we show that the proposed methods satisfy the discrete form of the stochastic energy linear growth property and preserve the multi-symplectic structure on the discrete level. Optimal error estimate of the semi-discrete DG method is also analyze…
We present a novel postprocessing technique for a discontinuous Galerkin (DG) discretization of time-dependent Maxwell's equations that we couple with an explicit Runge-Kutta time-marching scheme. The postprocessed electromagnetic field converges one order faster than the unprocessed solution in the H(curl)-norm. The proposed approach is local, in the sense that the enhanced solution is computed independently in each cell of the computational mesh, and at each time step of interest. As a result, it is inexpensive to compute, especially if the region of interest is localized, either in time or …
This paper considers the discontinuous Galerkin (DG) methods for solving the Vlasov-Maxwell (VM) system, a fundamental model for collisionless magnetized plasma. The DG methods provide accurate numerical description with conservation and stability properties. However, to resolve the high dimensional probability distribution function, the computational cost is the main bottleneck even for modern-day supercomputers. This work studies the applicability of a post-processing technique to the DG solution to enhance its accuracy and resolution for the VM system. In particular, we prove the superconve…
An error analysis of a mixed discontinuous Galerkin (DG) method with Brezzi numerical flux for the time-harmonic Maxwell equations with minimal smoothness requirements is presented. The key difficulty in the error analysis for the DG method is that the tangential or normal trace of the exact solution is not well-defined on the mesh faces of the computational mesh. We overcome this difficulty by two steps. First, we employ a lifting operator to replace the integrals of the tangential/normal traces on mesh faces by volume integrals. Second, optimal convergence rates are proven by using smoothed …
This paper proposes a fully discrete method called the symplectic dG full discretization for stochastic Maxwell equations driven by additive noises, based on a stochastic symplectic method in time and a discontinuous Galerkin (dG) method with the upwind fluxes in space. A priori $H^k$-regularity ($k\in\{1,2\}$) estimates for the solution of stochastic Maxwell equations are presented, which have not been reported before to the best of our knowledge. These $H^k$-regularities are vital to make the assumptions of the mean-square convergence analysis on the initial fields, the noise and the medium …
Numerical analysis of the second-order time-dependent saddle point Maxwell system via a parameter-free discontinuous Galerkin method: The first optimal ${\bf L}^{2}$-norm error estimates
Indian Institute of Technology Guwahati（印度理工学院古瓦哈提分校）
An unconditionally convergent discontinuous Galerkin neural network method activated by plane waves for Helmholtz equation and Maxwell's equations
College of Mathematics and Systems Science, Shandong University of Science and Technology（山东科技大学数学与系统科学学院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
A $p$-adaptive discontinuous Galerkin time-domain method is developed to obtain high-order solutions to electromagnetic scattering problems. A novel feature of the proposed method is the use of divergence error to drive the $p$-adaptive method. The nature of divergence error is explored and that it is a direct consequence of the act of discretization is established. Its relation with relative truncation error is formed which enables the use of divergence error as an inexpensive proxy to truncation error. Divergence error is used as an indicator to dynamically identify and assign spatial operat…
A Discontinuous Galerkin (DG) Finite Element Method (FEM) approach for the 3D time dependent Maxwell equations in unbounded domains is presented. The method (implemented in the FEM library NGsolve) is based on the covariant transformation of a modal orthogonal polynomial basis, originally defined on a reference simplex. The approach leads to an explicit time stepping scheme for which the mass matrix to be inverted is at most $d\!\times\!d$ block-diagonal (in $d\!=\!2,3$ spatial dimensions) while the matrix which discretizes the curl operators on the right-hand side of the system is a small ref…
We propose high-order FDTD schemes based on the Correction Function Method (CFM) for Maxwell's interface problems with discontinuous coefficients and complex interfaces. The key idea of the CFM is to model the correction function near an interface to retain the order of a finite difference approximation. For this, we solve a system of PDEs based on the original problem by minimizing an energy functional. The CFM is applied to the standard Yee scheme and a fourth-order FDTD scheme. The proposed CFM-FDTD schemes are verified in 2-D using the transverse magnetic mode (TM$_z$). Numerical examples …
In this work, we investigate the propagation of electromagnetic waves in the Cole-Cole dispersive medium by using the discontinuous Galerkin (DG) method to solve the coupled time-domain Maxwell's equations and polarization equation. We define a new and sharpened total energy function for the Cole-Cole model, which better describes the behaviors of the energy than what is available in the current literature. A major theme in the time-domain numerical modeling of this problem has been tackling the difficulty of handling the nonlocal term involved in the time-domain polarization equation. Based o…
The inverse of the stiffness matrix of the time-harmonic Maxwell equation with perfectly conducting boundary conditions is approximated in the blockwise low-rank format of ${\mathcal H}$-matrices. We prove that root exponential convergence in the block rank can be achieved if the block structure conforms to a standard admissibility criterion.
A priori and a posteriori analysis of the discontinuous Galerkin approximation of the time-harmonic Maxwell's equations under minimal regularity assumptions
Dispersion Analysis of Finite Difference and Discontinuous Galerkin Schemes for Maxwell's Equations in Linear Lorentz Media
University of Science and Technology of China（中国科学技术大学）；King Mongkut’s Institute of Technology Ladkrabang（拉加邦国王科技大学）；Oregon State University（俄勒冈州立大学）；Michigan State University（密歇根州立大学）；Rensselaer Polytechnic Institute（伦斯勒理工学院）
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…
A Trefftz Discontinuous Galerkin Method for Time Harmonic Waves with Generalized Impedance Boundary Conditions
Institute for Mathematics and its Applications, College of Science and Engineering（数学及其应用研究所，科学与工程学院）；Department of Mathematical Sciences, University of Delaware（特拉华大学数学科学系）；Departamento de Matemáticas, Universidad de Oviedo（奥维耶多大学数学系）
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）
We consider time-harmonic Maxwell's equations set in an heterogeneous medium with perfectly conducting boundary conditions. Given a divergence-free right-hand side lying in $L^2$, we provide a frequency-explicit approximability estimate measuring the difference between the corresponding solution and its best approximation by high-order Nédélec finite elements. Such an approximability estimate is crucial in both the a priori and a posteriori error analysis of finite element discretizations of Maxwell's equations, but the derivation is not trivial. Indeed, it is hard to take advantage of high-or…
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…
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 …
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…
We consider finite element discretizations of Maxwell's equations coupled with a non-local hydrodynamic Drude model that accurately accounts for electron motions in metallic nanostructures. Specifically, we focus on a posteriori error estimation and mesh adaptivity, which is of particular interest since the electromagnetic field usually exhibits strongly localized features near the interface between metals and their surrounding media. We propose a novel residual-based error estimator that is shown to be reliable and efficient. We also present a set of numerical examples where the estimator dri…
We propose an unfitted finite element method for numerically solving the time-harmonic Maxwell equations on a smooth domain. The model problem involves a Lagrangian multiplier to relax the divergence constraint of the vector unknown. The embedded boundary of the domain is allowed to cut through the background mesh arbitrarily. The unfitted scheme is based on a mixed interior penalty formulation, where Nitsche penalty method is applied to enforce the boundary condition in a weak sense, and a penalty stabilization technique is adopted based on a local direct extension operator to ensure the stab…
We survey finite element methods for approximating the time harmonic Maxwell equations. We concentrate on comparing error estimates for problems with spatially varying coefficients. For the conforming edge finite element methods, such estimates allow, at least, piecewise smooth coefficients. But for Discontinuous Galerkin (DG) methods, the state of the art of error analysis is less advanced (we consider three DG families of methods: Interior Penalty type, Hybridizable DG, and Trefftz type methods). Nevertheless, DG methods offer significant potential advantages compared to conforming methods.
