Time-harmonic Maxwell Equations Research Articles

On the Regularity of Time-Harmonic Maxwell Equations with Impedance Boundary Conditions

On the Regularity of Time-Harmonic Maxwell Equations with Impedance Boundary Conditions

On a critical time-harmonic Maxwell equation in nonlocal media

In this paper, we study the existence of solutions for a critical time–harmonic Maxwell equation in nonlocal media \[ \begin{cases} \nabla\times(\nabla\times u)+\lambda u=\left(I_{\alpha}\ast|u|^{2^{{\ast}}_{\alpha}}\right)|u|^{2^{{\ast}}_{\alpha}-2}u & \mathrm{in}\ \Omega,\\ \nu\times u=0 & \mathrm{on}\ \partial\Omega, \end{cases} \] where $\Omega \subset \mathbb {R}^{3}$ is a bounded domain, either convex or with $\mathcal {C}^{1,1}$ boundary, $\nu$ is the exterior normal, $\lambda <0$ is a real parameter, $2^{\ast }_{\alpha }=3+\alpha$ with $0<\alpha <3$ is the upper critical exponent due to the Hardy–Littlewood–Sobolev inequality. By introducing some suitable Coulomb spaces involving curl operator $W^{\alpha,2^{\ast }_{\alpha }}_{0}(\mathrm {curl};\Omega )$ , we are able to obtain the ground state solutions of the curl–curl equation via the method of constraining Nehari–Pankov manifold. Correspondingly, some sharp constants of the Sobolev-like inequalities with curl operator are obtained by a nonlocal version of the concentration–compactness principle.

A PWDG method for the Maxwell system in anisotropic media with piecewise constant coefficient matrix

In this paper we are concerned with plane wave discontinuous Galerkin (PWDG) methods for time-harmonic Maxwell equations in three-dimensional anisotropic media, for which the coefficients of the equations are piecewise constant symmetric matrices, where each constant symmetric matrix is defined on a medium (subdomain). By using suitable scaling transformations and coordinate (complex) transformations on every subdomain, the original Maxwell equation in anisotropic media is transformed into a Maxwell equation in isotropic media occupying a union domain of specific subdomains of complex Euclidean space. Based on these transformations, we define anisotropic plane wave basis functions and discretize the considered Maxwell equations by PWDG method with the proposed plane wave basis functions. We derive error estimates of the resulting approximate solutions, and further introduce a practically feasible local hp-refinement algorithm, which substantially improves accuracies of the approximate solutions. Numerical results indicate that the approximate solutions generated by the proposed PWDG methods possess high accuracy for the case of strong discontinuity media.

Wavenumber-Explicit hp-FEM Analysis for Maxwell’s Equations with Impedance Boundary Conditions

AbstractThe time-harmonic Maxwell equations at high wavenumberkin domains with an analytic boundary and impedance boundary conditions are considered. A wavenumber-explicit stability and regularity theory is developed that decomposes the solution into a part with finite Sobolev regularity that is controlled uniformly inkand an analytic part. Using this regularity, quasi-optimality of the Galerkin discretization based on Nédélec elements of orderpon a mesh with mesh sizehis shown under thek-explicit scale resolution condition that (a)kh/pis sufficient small and (b)$$p/\ln k$$p/lnkis bounded from below.

A three-dimensional Petrov-Galerkin finite element interface method for solving inhomogeneous anisotropic Maxwell's equations in irregular regions

In this paper, a three-dimensional Petrov-Galerkin finite element interface (3D PGFEI) method with non-body-fitted grids is proposed for solving the inhomogeneous anisotropic time-harmonic Maxwell's equations in irregular regions. We derive the weak variational form of the anisotropic Maxwell interface problem. The irregular region is discretized by uniform tetrahedral grids while the media interface is not aligned with the grids. The level-set functions are applied to capture the discontinuity on the anisotropic media interface. Meanwhile, we construct the special basis functions according to the jump conditions across the inhomogeneous interface, where the permittivity and permeability are discontinuous three-dimensional tensors. The resulting linear system is a banded sparse matrix and demands fewer memory resources than dense matrix. Numerical experiments illustrate the efficiency and accuracy of the proposed 3D PGFEI method and the solutions can achieve second-order accuracy in terms of the relative L∞(Ω) and relative L2(Ω) error norms.

Explicit bounds for the high-frequency time-harmonic Maxwell equations in heterogeneous media

We consider the time-harmonic Maxwell equations posed in R3. We prove a priori bounds on the solution for L∞ coefficients ϵ and μ satisfying certain monotonicity properties, with these bounds valid for arbitrarily-large frequency, and explicit in the frequency and properties of ϵ and μ. The class of coefficients covered includes (i) certain ϵ and μ for which well-posedness of the time-harmonic Maxwell equations had not previously been proved, and (ii) scattering by a penetrable C0 star-shaped obstacle where ϵ and μ are smaller inside the obstacle than outside. In this latter setting, the bounds are uniform across all such obstacles, and the first sharp frequency-explicit bounds for this problem at high-frequency.

Adaptive edge finite element method and numerical design for metasurface cloak

Metasurfaces are metamaterials with two-dimensional (2D) or planar structures. Compared to traditional metamaterials, metasurfaces have the advantage of being lightweight, easy to control and simple to design. This paper outlines the phase matching principle and constitutive parameter theory of electromagnetic (EM) metasurfaces, as well as the constitutive parameter analysis and model equations for metasurface cloaks. Due to the singularity of the metasurface cloak, an Adaptive Edge Finite Element Method (AEFEM) is developed in this paper. In addition, the convergence theory of the AEFEM for the time-harmonic Maxwell's equations with one sign-changing coefficient is established under the T-coercivity assumption. Numerical simulations of the arc-shaped metasurface cloak have confirmed the accuracy of our model and numerical theory. Furthermore, we have used our numerical method to design a triangular-shaped metasurface cloak with fewer singularities.

EMFEM: A parallel 3D modeling code for frequency-domain electromagnetic method using goal-oriented adaptive finite element method

We have developed an open-source parallel 3D forward modeling code called EMFEM for the frequency-domain electromagnetic method (FDEM). This code is based on the adaptive finite element method and uses the total-field approach to solve the time-harmonic Maxwell equations. The use of tetrahedral meshes enables modeling of problems with complex geometry and topography. We utilize an iterative solver and an optimal block-diagonal preconditioner and auxiliary Maxwell solver to accelerate the convergence rate. We also use goal-oriented error estimation to adaptively refine the mesh and improve the accuracy of the solution. In addition, we develop a parallel algorithm based on the mesh partitioning approach. The effectiveness of the adaptive mesh refinement is demonstrated through a 1D layered model for marine controlled-source electromagnetic (CSEM) problems. We further present two 3D models for CSEM and magnetotelluric (MT) problems to study the scalability of the parallel algorithm, and the accuracy of the solutions is cross-validated with other open-source codes. The code is implemented in C++ and follows the modular design principle, making it easy to extend and maintain. The code is capable of modeling large-scale 3D problems and utilizing high-performance computing platforms. Furthermore, as the code is freely available, it can serve as a cross-validation tool for other 3D modeling codes and can also be utilized to implement 3D inversion routines.

Domain Decomposition with Neural Network Interface Approximations for time-harmonic Maxwell’s equations with different wave numbers

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 some numerical experiments with low frequencies. Therein the new approach is compared to a classical domain decomposition method. Moreover, we highlight current challenges of training and testing with different wave numbers and we provide information on the behaviour of the neural-network, such as convergence of the loss function, and different activation functions.

Ground state solution of weakly coupled time-harmonic Maxwell equations

Ground state solution of weakly coupled time-harmonic Maxwell equations

An OSRC Preconditioner for the EFIE

The electric field integral equation (EFIE) is a well-established tool to solve electromagnetic scattering problems. However, the development of efficient and easy-to-implement preconditioners remains an active research area. In recent years, operator preconditioning approaches have become popular for the EFIE, where the electric field boundary integral operator is regularized by multiplication with another convenient operator. A particularly intriguing choice is the exact magnetic-to-electric (MtE) operator as a regularizer. But, evaluating this operator is as expensive as solving the original EFIE. In work by El Bouajaji, Antoine, and Geuzaine, approximate local MtE surface operators for the time-harmonic Maxwell equation were proposed. These can be efficiently evaluated through the solution of sparse problems. This article demonstrates the preconditioning properties of these approximate MtE operators for the EFIE. The implementation is described, and a number of numerical comparisons against other preconditioning techniques for the EFIE are presented to demonstrate the effectiveness of this new technique.

Determination of piecewise homogeneous sources for elastic and electromagnetic waves

This paper is concerned with inverse source problems for the time-harmonic elastic wave equations and Maxwell's equations with a single boundary measurement at a fixed frequency. We show the uniqueness and a Lipschitz-type stability estimate under the assumption that the source function is piecewise constant on a domain which is made of a union of disjoint convex polyhedral subdomains.

A Controllability Method for Maxwell's Equations

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 nonintrusive implementation easily integrated into existing software. Moreover, the proposed algorithm automatically inherits the parallelism, scalability, and low memory footprint of the underlying time-domain solver. Since the time-periodic solution obtained by minimization is not necessarily unique, we apply a cheap postprocessing filtering procedure which recovers the time-harmonic solution from any minimizer. Finally, we present a series of numerical examples which show that our algorithm greatly speeds up the convergence toward the desired time-harmonic solution when compared to simply running the time-marching code until the time-harmonic regime is eventually reached.

The vector Helmholtz equation revisited: Inverse obstacle scattering

The vector Helmholtz equation, from a mathematical point of view, provides a generalization of the time-harmonic Maxwell equations for the propagation of time-harmonic electromagnetic waves. After reviewing some classic results on the two main exterior boundary value problems for the vector Helmholtz equation, i.e., the so-called electric boundary condition and the magnetic boundary condition, we prove reciprocity results for scattering of plane waves and point sources. Then we make use of them for obtaining uniqueness results for the inverse obstacle scattering problem to determine the shape of the scatterer from knowing the far field pattern of the scattered waves and results on the so-called far field operator for the two scattering problems. These results are generalizations of corresponding results for the Maxwell equations and analogues to results for the Dirichlet and Neumann boundary condition for the scalar Helmholtz equation. We also briefly consider the extension of the so-called DB boundary condition from the Maxwell equations to the vector Helmholtz equation.

Sparse Polynomial Approximations for Affine Parametric Saddle Point Problems

In this work we study convergence properties of sparse polynomial approximations for a class of affine parametric saddle point problems, including the Stokes equations for viscous incompressible flow, mixed formulation of diffusion equations for groundwater flow, time-harmonic Maxwell equations for electromagnetics, etc. Due to the lack of knowledge or intrinsic randomness, the (viscosity, diffusivity, permeability, permittivity, etc.) coefficients of such problems are uncertain and can often be represented or approximated by high- or countably infinite-dimensional random parameters equipped with suitable probability distributions, and the coefficients affinely depend on a series of either globally or locally supported basis functions, e.g., Karhunen–Loève expansion, piecewise polynomials, or adaptive wavelet approximations. We consider sparse polynomial approximations of the parametric solutions, in particular sparse Taylor approximations, and study their convergence properties for these parametric problems. Under suitable sparsity assumptions on the parametrization of the random coefficients, we show the algebraic convergence rates O(N−r) for the sparse polynomial approximations of the parametric solutions based on the results for affine parametric elliptic PDEs (Cohen, A. et al.: Anal. Appl. 9, 11–47, 2011), (Bachmayr, M., et al.: ESAIM Math. Model. Numer. Anal. 51, 321–339, 2017), (Cohen, A., DeVore, R.: Acta Numer. 24, 1–159, 2015), (Chkifa, A., et al.: J. Math. Pures Appl. 103, 400–428, 2015), (Chkifa, A., et al.: ESAIM Math. Model. Numer. Anal. 47, 253–280, 2013), (Cohen, A., Migliorati, G.: Contemp. Comput. Math., 233–282, 2018), with the rate r depending only on a sparsity parameter in the parametrization, not on the number of active parameter dimensions or the number of polynomial terms N. We note that parametric saddle point problems were considered in (Cohen, A., DeVore, R.: Acta Numer. 24, 1–159, 2015, Section 2.2) with the anticipation that the same results on the approximation of the solution map obtained for elliptic PDEs can be extended to more general saddle point problems. In this paper, we consider a general formulation of saddle point problems, different from that presented in (Cohen, A., DeVore, R.: Acta Numer. 24, 1–159, 2015, Section 2.2), and obtain convergence rates for the two variables, e.g., velocity and pressure in Stokes equations, which are different for the case of locally supported basis functions.

On the approximation of electromagnetic fields by edge finite elements—Part 4: analysis of the model with one sign-changing coefficient

In electromagnetism, in the presence of a negative material surrounded by a classical material, the electric permittivity, and possibly the magnetic permeability, can exhibit a sign-change at the interface. In this setting, the study of electromagnetic phenomena is a challenging topic. We focus on the time-harmonic Maxwell equations in a bounded set \(\varOmega \) of \(\mathbb {R}^3\), and more precisely on the numerical approximation of the electromagnetic fields by edge finite elements. Special attention is paid to low-regularity solutions, in terms of the Sobolev scale \((\varvec{H}^{\texttt {s}}(\varOmega ))_{\texttt {s}>0}\). With the help of T-coercivity, we address the case of one sign-changing coefficient, both for the model itself, and for its discrete version. Optimal a priori error estimates are derived.

{mathcal{H}}-matrix approximability of inverses of FEM matrices for the time-harmonic Maxwell equations

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. Under a technical assumption on the mesh, we prove that root exponential convergence in the block rank can be achieved, if the block structure conforms to a standard admissibility criterion.

A weak Galerkin finite element method for indefinite time-harmonic Maxwell equations

In this paper, we intend to develop a weak Galerkin (WG) finite element method for solving the indefinite time-harmonic Maxwell equations. Firstly, by using an analogy of Ga˚rding inequality and proving a posteriori estimate about the error, we prove the well-posedness of the WG method. Then, by deducing an error equation, we achieve optimal a priori error estimates in both the energy norm and the L2 norm. Finally, we carry out some numerical experiments to confirm the theoretical conclusions.

Analysis of a mixed discontinuous Galerkin method for the time-harmonic Maxwell equations with minimal smoothness requirements

AbstractAn error analysis of a mixed discontinuous Galerkin (DG) method with lifting operators as numerical fluxes 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 due to the low regularity the tangential trace of the exact solution is not well defined on the faces of the computational mesh. This difficulty is addressed by adopting the face-to-cell lifting introduced by Ern & Guermond (2021, Quasi-optimal nonconforming approximation of elliptic PDEs with contrasted coefficients and $H^{1+r}$, $r>0$, regularity. Found. Comput. Math., 1–36). To obtain optimal local interpolation estimates, we introduce Scott–Zhang-type interpolations that are well defined for $H(\textrm {curl})$ and $H(\textrm {div})$ functions with minimal regularity requirements. As a by-product of penalizing the lifting of the tangential jumps, an explicit and easily computable stabilization parameter is given.

Frequency-Explicit A Posteriori Error Estimates for Finite Element Discretizations of Maxwell's Equations

We consider residual-based a posteriori error estimators for Galerkin-type discretizations of time-harmonic Maxwell's equations. We focus on configurations where the frequency is high, or close to a resonance frequency, and derive reliability and efficiency estimates. In contrast to previous related works, our estimates are frequency-explicit. In particular, our key contribution is to show that even if the constants appearing in the reliability and efficiency estimates may blow up on coarse meshes, they become independent of the frequency for sufficiently refined meshes. Such results were previously known for the Helmholtz equation describing scalar wave propagation problems and we show that they naturally extend, at the price of many technicalities in the proofs, to Maxwell's equations. Our mathematical analysis is performed in the 3D case, and covers conforming N\'ed\'elec discretizations of the first and second family, as well as first-order (and hybridizable) discontinuous Galerkin schemes. We also present numerical experiments in the 2D case, where Maxwell's equations are discretized with N\'ed\'elec elements of the first family. These illustrating examples perfectly fit our key theoretical findings, and suggest that our estimates are sharp.