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1.
By a general argument, it is shown that Maxwell–Herglotz‐fields are dense (with respect to the C(Ω)‐topology) in the space of all solutions to Maxwell's equations in Ω. This is used to provide corresponding approximation results in global spaces (e.g. in L2‐Sobolev‐spaces Hm(Ω)) and for boundary data. Proofs are given within the framework of generalized Maxwell's equations using differential forms. Copyright © 2004 John Wiley & Sons, Ltd.  相似文献   

2.
This paper is concerned with new energy analysis of the two dimensional Maxwell's equations and the symmetric energy‐conserved splitting finite difference time domain (EC‐S‐FDTD) method with the periodic boundary (PB) condition. New energy identities of the Maxwell's equations in terms of H1 and H2 norms are proposed and interpreted by considering the physical meanings of the H1 and H2 semi‐norms in the identities. It is found from these new identities that the first and second curls of the electromagnetic fields are conserved in terms their magnitudes. By the energy methods, the numerical energy identities of the symmetric EC‐S‐FDTD method are derived and shown to converge to the continuous energy identities of the Maxwell's equations. This proves that the symmetric EC‐S‐FDTD scheme is unconditionally stable and energy conserved in the discrete H1 and H2 norms. Also by the energy methods, it is proved that the symmetric EC‐S‐FDTD method with PB condition is of second order (super) convergence in the discrete H1 and H2 norms. Numerical experiments are carried out and confirm the analysis on energy conservation, stability and super convergence.  相似文献   

3.
We study an induction hardening model described by Maxwell's equations coupled with a heat equation. The magnetic induction field is assumed a nonlinear constitutional relation and the electric conductivity is temperature‐dependent. The Tψ method is to transform Maxwell's equations to the vector–scalar potential formulations and to solve the potentials by means of the finite element method. In this article, we present a fully discrete Tψ finite element scheme for this nonlinear coupled problem and discuss its solvability. We prove that the discrete solution converges to a weak solution of the continuous problem. Finally, we conclude with two numerical experiments for the coupled system.  相似文献   

4.
In this article, we consider the time‐dependent Maxwell's equations modeling wave propagation in metamaterials. One‐order higher global superclose results in the L2 norm are proved for several semidiscrete and fully discrete schemes developed for solving this model using nonuniform cubic and rectangular edge elements. Furthermore, L superconvergence at element centers is proved for the lowest order rectangular edge element. To our best knowledge, such pointwise superconvergence result and its proof are original, and we are unaware of any other publications on this issue. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential 2011  相似文献   

5.
6.
This paper is concerned with the structure of the singular and regular parts of the solution of time‐harmonic Maxwell's equations in polygonal plane domains and their effective numerical treatment. The asymptotic behaviour of the solution near corner points of the domain is studied by means of discrete Fourier transformation and it is proved that the solution of the boundary value problem does not belong locally to H2 when the boundary of the domain has non‐acute angles. A splitting of the solution into a regular part belonging to the space H2, and an explicitly described singular part is presented. For the numerical treatment of the boundary value problem, we propose a finite element discretization which combines local mesh grading and the singular field methods and derive a priori error estimates that show optimal convergence as known for the classical finite element method for problems with regular solutions. Copyright © 2006 John Wiley & Sons, Ltd.  相似文献   

7.
Let u be a vector field on a bounded Lipschitz domain in ?3, and let u together with its divergence and curl be square integrable. If either the normal or the tangential component of u is square integrable over the boundary, then u belongs to the Sobolev space H1/2 on the domain. This result gives a simple explanation for known results on the compact embedding of the space of solutions of Maxwell's equations on Lipschitz domains into L2.  相似文献   

8.
Kovats Jay 《偏微分方程通讯》2013,38(11-12):1911-1927
Abstract

We investigate transmission problems with strongly Lipschitz interfaces for the Dirac equation by establishing spectral estimates on an associated boundary singular integral operator, the rotation operator. Using Rellich estimates we obtain angular spectral estimates on both the essential and full spectrum for general bi-oblique transmission problems. Specializing to the normal transmission problem, we investigate transmission problems for Maxwell's equations using a nilpotent exterior/interior derivativeoperator. The fundamental commutation properties for this operator with the two basic reflection operators are proved. We show how the L 2spectral estimates are inherited for the domain of the exterior/interior derivative operator and prove some complementary eigenvalue estimates. Finally we use a general algebraic theorem to prove a regularity property needed for Maxwell's equations.  相似文献   

9.
In this paper we consider the inverse backscattering problem for Maxwell's equations in a non-magnetic inhomogeneous medium, i.e. the magnetic permeability is a fixed constant. We show that the electric permittivity ε is uniquely determined by the trace of the backscattering kernel S(s, −θ, θ) for all s∈ℝ, θ∈ S 2 provided that it is a priori close to a constant. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.  相似文献   

10.
We shortly revisit the Heterogeneous Multiscale Method (HMM) for the time-harmonic Maxwell's equations in locally periodic media as introduced in [1]. The optimal a priori bounds in the H (curl) and H−1 norm predicted theoretically are justified by a numerical example in this contribution. The setting consists of a periodic inverse permeability, inspired by [2], and as reference solution a computation on a fine (well-resolved) grid is used. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

11.
The treatment of boundary value problems for Helmholtz equation and for the time harmonic Maxwell's equations by boundary integral equations leads to integral equations of the second kind which are uniquely solvable for small positive frequencies λ. However, the integral equations obtained in the limiting case λ = 0 which are related to boundary value problems of potential theory in general are not uniquely solvable since the corresponding boundary value problems are not. By first considering in a general setting of a Banach space X the limiting behaviour of solutions ?λ to the equation ?λ – K λ ? λ = fλ as λ → 0 where {Kλ: XX, λ ∈ (0,α)}, α > 0, denotes a family of compact linear operators such that I - Kλ (I identity) is bijective for λ∈(0,α) whilst I - K0 is not and ‖ KλK0‖ →, 0, ‖fλf0‖ → 0, λ → 0, and then applying the results to the boundary integral operators, the limiting behaviour of the integral equations is considered. Thus, the results obtained by Mac Camey for the Helmholtz equation are extended to the case of non-connected boundaries and Werner's results on the integral equations for the Maxwell's equations are extended to the case of multiply connected boundaries.  相似文献   

12.
In this article, we study the Drude models of Maxwell's equations in three‐dimensional metamaterials. We derive new global energy‐tracking identities for the three dimensional electromagnetic problems in the Drude metamaterials, which describe the invariance of global electromagnetic energy in variation forms. We propose the time second‐order global energy‐tracking splitting FDTD schemes for the Drude model in three dimensions. The significant feature is that the developed schemes are global energy‐preserving, unconditionally stable, second‐order accurate both in time and space, and computationally efficient. We rigorously prove that the new schemes satisfy these energy‐tracking identities in the discrete form and the discrete variation form and are unconditionally stable. We prove that the schemes in metamaterials are second order both in time and space. The superconvergence of the schemes in the discrete H1 norm is further obtained to be second order both in time and space. Their approximations of divergence‐free are also analyzed to have second‐order accuracy both in time and space. Numerical experiments confirm our theoretical analysis results. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 763–785, 2017  相似文献   

13.
Let U be a connected, closed, bounded region in ℝ3 with smooth boundary 𝛛U. Consider Maxwell's equations on U for smooth fields and smooth sources, which are time harmonic, i.e., the fields E,B,H,D and the current density J and charge density ρ on U temporally varying as eiωt. We assume ω ≠ 0; when it is real, ω is the frequency of the pure oscillation; when ω is not real, one has exponentially decreasing or increasing oscillatory fields. We treat very general electric permittivity ɛω and magnetic permeability μω, which may both depend upon ω. Both can be arbitrary, time-independent, smoothly varying with position and described by complex Hermitian tensors subject only to weak positivity conditions specified below at each point of U. In this paper necessary and sufficient conditions are found for the tangential components, ET or HT on 𝛛U, to be realized by solutions of Maxwell's equations on all of U with these given sources J, ρ in the time-harmonic realm with ω ≠ 0. Also, all solutions E,H on U with specified ET, HT, respectively, satisfying the necessary finite-dimensional conditions are derived by an effective natural algorithm. In the electric ET case, the positivity condition is this: the complex Hermitian matrix μω(p) is to be positive definite while only the real part of ɛω(p), i.e., Re(ɛω(p)), necessarily real symmetric, need be positive definite. In the magnetic-type HT case, the two roles are reversed. These positivity conditions are more general than those commonly considered. In the final section Maxwell's equations and all the present results are reformulated yet more generally in a context of compact, smooth, Riemannian manifolds with boundary. Sample examples include periodic lattices and wave guides. © 2019 Wiley Periodicals, Inc.  相似文献   

14.
In this paper, we analyze the energy‐conserved splitting finite‐difference time‐domain (FDTD) scheme for variable coefficient Maxwell's equations in two‐dimensional disk domains. The approach is energy‐conserved, unconditionally stable, and effective. We strictly prove that the EC‐S‐FDTD scheme for the variable coefficient Maxwell's equations in disk domains is of second order accuracy both in time and space. It is also strictly proved that the scheme is energy‐conserved, and the discrete divergence‐free is of second order convergence. Numerical experiments confirm the theoretical results, and practical test is simulated as well to demonstrate the efficiency of the proposed EC‐S‐FDTD scheme. Copyright © 2015 John Wiley & Sons, Ltd.  相似文献   

15.
In this article, we consider the time‐dependent Maxwell's equations in a bounded domain when dispersive media are involved. The Crank‐Nicolson scheme is developed to approximate the electric field equation by Nedelec edge elements and is proved to be optimal convergent in energy norm. The analysis is carried out for Debye medium, but the same results hold true for other dispersive media such as plasma and Lorentz medium. Furthermore, our analysis extends straightforward to cases when a dispersive medium and a simple medium (such as air) are coupled. Mathematics Subject Classification (2000): 65N30, 35L15, 78‐08. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008  相似文献   

16.
In this paper, we study a numerical scheme to solve coupled Maxwell's equations with a nonlinear conductivity. This model plays an important role in the study of type‐II superconductors. The approximation scheme is based on backward Euler discretization in time and mixed conforming finite elements in space. We will prove convergence of this scheme to the unique weak solution of the problem and develop the corresponding error estimates. As a next step, we study the stability of the scheme in the quasi‐static limit ? → 0 and present the corresponding convergence rate. Finally, we support the theory by several numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.  相似文献   

17.
We use a bivariate spline method to solve the time evolution Navier‐Stokes equations numerically. The bivariate splines we use in this article are in the spline space of smoothness r and degree 3r over triangulated quadrangulations. The stream function formulation for the Navier‐Stokes equations is employed. Galerkin's method is applied to discretize the space variables of the nonlinear fourth‐order equation, Crank‐Nicholson's method is applied to discretize the time variable, and Newton's iterative method is then used to solve the resulting nonlinear system. We show the existence and uniqueness of the weak solution in L2(0, T; H2(Ω)) ∩ L(0, T; H1(Ω)) of the 2D nonlinear fourth‐order problem and give an estimate of how fast the numerical solution converges to the weak solution. The C1 cubic splines are implemented in MATLAB for solving the Navier‐Stokes equations numerically. Our numerical experiments show that the method is effective and efficient. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 776–827, 2003.  相似文献   

18.
Maxwell's boundary value problem for the time-harmonic case in a smooth, bounded domain G of R 2 is considered. The optimal asymptotic L2(G) and H1(G)-error estimates 0(h2) and 0(h) resp, are derived for a piecewise linear finite element solution.  相似文献   

19.
20.
Shumin Li 《Applicable analysis》2013,92(11):2335-2356
In this article, we consider Maxwell's equations in an isotropic, inhomogeneous and non-stationary medium. We discuss an inverse problem of determining the t-independent components of the coefficients ?, μ in the constitutive relations from a finite number of interior measurements. We prove a Lipschitz stability estimate for the inverse problem by applying the argument on the basis of Carleman estimate.  相似文献   

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