| The study of electromagnetic scattering and inverse scattering problems is an important topic in mathematical physics. There has been a lot of work in this region. For the references, we refer to [6], [8], [14], [16], [23], [24], [52].Recently, electromagnetic scattering and inverse scattering problems have been paid attention. Ammari and Nedelec, Athanasiadis, Ammari and Bao, Gerlach have studied some concerned problems.In chiral medium, time-harmonic electromagnetic fields are governed by the Maxwell equationsand the Drude-Born-Fedorov constitutive equationswhere x G R3, E,H,D and B are the electric field, the magnetic field, the electric and magnetic displacement vectors in R3, respectively; e(x), u(x) and B{x) are the electric permittivity, the magnetic permeability and the chirality admittance, respectively.Because of complexity of differential equations (e.g. the coupling of electric field and magnetic field) and the character of computation domain (e.g.unboundedness), there are a lot of difficulties in computational methods. To find effective computational methods is an important subject.In this paper, theoretical analysis and numerical methods of electromagnetic scattering and inverse scattering problems for chiral media are discussed.Firstly, we study electromagnetic scattering from periodic chiral structures. Given incident plane waves Ex = seiq.x and Hi = peiq.x such thatto find bounded quasi-periodic solutions E and H of the following equations (derived from equations(1)-(2))where are real valued functions which satisfyfor fixed positive A, b and sufficiently small > 0, positive constants We assume that the electromagnetic fields is invariant in the x3 direction, then the x1- and x2- components of electromagnetic fields can be expressed in terms of the x3-components. We get the boundary conditions easily by the boundedness and periodicity of electromagnetic fields. Therefore, we reduce the original problem to a two-dimensional scattering problem: to find e, h € Hp1() such that is the subset of all functions in H1() which are the restrictions to of the functions in if,1 (R x (-b, b)) and A-periodic on direction, the operators Tj(j = 1,2) are defined byFor convenience, we drop the subscript a from ea and ha and the subscript p from H1p() without confusion. Prom now on, we will use to denote the domain Integrating by parts and using the boundary conditions , we haveDenote the left hand sides of (10), (11) by a(e,p), b(h,q), respectively. Then, the weak form of the above boundary value problem (6)-(9) is to find such thatwhereBy using the meromorphic Fredholm theorem, we investigate the well-posedness of problem(12)-(14) and prove existence and uniqueness of the weak quasi-periodic solutions.Theorem 1 For all but possibly a discrete set of frequencies w, the variational problem (12)-(14) admits a unique solution u in H1( ) x H1()Then, we establish uniform error estimate for the finite element method. Theorem 2 Suppose that (12)-(14) has a unique solution u H1( ) x H1 () for each f . Then for any given t > 0, there exists an h0 = h0() such that, for Moreover, if , there exists an such that, for 0 < h< h1,It is obvious that we can not compute the nonlocal operators Tj(j = 1,2) from the infinite series expansions. Thus, we truncate the boundary operators and obtain appropriate error estimates.Theorem 3 Suppose that problem (12)-(14) has a unique solution u . Then for any given e > 0 and sufficiently large integer N, there exists an such that, for 0 < h < h0, the truncated problem attains a unique solution Moreover, the following estimates hold:In addition, , there exists an h1 = h1(e) such that, for 0 < h< h1,where C depends on but is independent of h, N, and u.We also study the energy distribution for the diffraction problem and establish energy conservation formulation. We use et,hr and et,ht to denote the energy of reflected and transmitted waves, respectively (the details can be found in ?.6). Then we have the followin...
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