Discontinuous Galerkin Approximations For Computing Band Gaps In Photonic Crystals

In this dissertation,We mainly study the spectral approximation theory and its applications for differential operators related to photonic crystals.We analyze discontinuous Galerkin finite element discretizations of the Maxwell equations with periodic coefficients.These equations are used to model the behavior of light in photonic crystals,which are materials containing a spatially periodic vari-ation of the refractive index commensurate with the wavelength of light.Depending on the geometry,material properties and lattice structure these materials exhibit a pho-tonic band gap in which light of certain frequencies is completely prohibited inside the photonic crystal.By Bloch/Floquet theory,this problem is equivalent to a modified Maxwell eigenvalue problem with periodic boundary conditions,which is discretized with a mixed discontinuous Galerkin(DG)formulation using modified Nedelec basis functions.We also investigate an alternative primal DG interior penalty formulation and compare this method with the mixed DG formulation.We prove the convergence for these DG methods.However,this kind convergence is pointwise convergence,and the correctness of numerical eigenvalues is equivalent to uniform convergence.To this end.we prove a discrete compactness property for the corresponding DG space.The convergence rate of the numerical eigenvalues is twice the minimum of the order of the polynomial basis functions and the regularity of the solution of the Maxwell equations.We present both 2D and 3D numerical examples to verify the convergence rate of the mixed DG method and demonstrate its application to computing the band structure of photonic crystals.When the properties of material for photonic crystals are nonlinear,nonlinear eigen-value problems will arise.We study the polynomial eigenvalue problems(PEP)pro-duced in rational models,for instance,Durde model and Lorentz model.The linearized operators that shares the same spectra with PEPs are possibly unbounded,which we can no longer use norms to describe the convergence for their corresponding numeri-cal operators.To overcome this difficulty,we introduce gaps to measure the distances between linear operators in finite dimensional spaces.With this new tool,we develop a spectral approximation theory for linear operators,and apply it to the PEPs.We also study the essential spectrum in PEPs,and prove that this kind of spectra is stable under relatively compact perturbations.Thus,only from theoretical analysis,we can obtain the exact values of essential spectrum of a complicate PEP.In complex plane,there are large number of similar numerical eigenvalues of corresponding discrete PEPs around essential spectrum.In computations,we can avoid these small regions,which can make the algorithms for algebra systems much more efficient without losing important infor-mation.