Existence, Uniqueness and Regularity for Solutions of the Conical Diffraction Problem

This paper is devoted to the analysis of two Helmholtz equations in ℝ2 coupled via quasiperiodic transmission conditions on a set of piecewise smooth interfaces. The solution of this system is quasi-periodic in one direction and satisfies outgoing wave conditions with respect to the other direction. It is shown that Maxwell's equations for the diffraction of a time-harmonic oblique incident plane wave by periodic interfaces can be reduced to problems of this kind. The analysis is based on a strongly elliptic variational formulation of the differential problem in a bounded periodic cell involving nonlocal boundary operators. We obtain existence and uniqueness results for solutions corresponding to electromagnetic fields with locally finite energy. Special attention is paid to the regularity and leading asymptotics of solutions near the edges of the interface

Application of Helmholtz/Hodge Decomposition to Finite Element Methods for Two-Dimensional Maxwell\u27s Equations

In this work we apply the two-dimensional Helmholtz/Hodge decomposition to develop new finite element schemes for two-dimensional Maxwell\u27s equations. We begin with the introduction of Maxwell\u27s equations and a brief survey of finite element methods for Maxwell\u27s equations. Then we review the related fundamentals in Chapter 2. In Chapter 3, we discuss the related vector function spaces and the Helmholtz/Hodge decomposition which are used in Chapter 4 and 5. The new results in this dissertation are presented in Chapter 4 and Chapter 5. In Chapter 4, we propose a new numerical approach for two-dimensional Maxwell\u27s equations that is based on the Helmholtz/Hodge decomposition for divergence-free vector fields. In this approach an approximate solution for Maxwell\u27s equations can be obtained by solving standard second order scalar elliptic boundary value problems. This new approach is illustrated by a P1 finite element method. In Chapter 5, we further extend the new approach described in Chapter 4 to the interface problem for Maxwell\u27s equations. We use the extraction formulas and multigrid method to overcome the low regularity of the solution for the Maxwell interface problem. The theoretical results obtained in this dissertation are confirmed by numerical experiments

Locally implicit discontinuous Galerkin method for time domain electromagnetics

In the recent years, there has been an increasing interest in discontinuous Galerkin time domain (DGTD) methods for the solution of the unsteady Maxwell equations modeling electromagnetic wave propagation. One of the main features of DGTD methods is their ability to deal with unstructured meshes which are particularly well suited to the discretization of the geometrical details and heterogeneous media that characterize realistic propagation problems. Such DGTD methods most often rely on explicit time integration schemes and lead to block diagonal mass matrices. However, explicit DGTD methods are also constrained by a stability condition that can be very restrictive on highly refined meshes and when the local approximation relies on high order polynomial interpolation. An implicit time integration scheme is a natural way to obtain a time domain method which is unconditionally stable but at the expense of the inversion of a global linear system at each time step. A more viable approach consists of applying an implicit time integration scheme locally in the refined regions of the mesh while preserving an explicit time scheme in the complementary part, resulting in an hybrid explicit–implicit (or locally implicit) time integration strategy. In this paper, we report on our recent efforts towards the development of such a hybrid explicit–implicit DGTD method for solving the time domain Maxwell equations on unstructured simplicial meshes. Numerical experiments for 3D propagation problems in homogeneous and heterogeneous media illustrate the possibilities of the method for simulations involving locally refined meshes

Mini-Workshop: Analytical and Numerical Treatment of Singularities in PDE

[no abstract available

Effective Cell-Centred Time-Domain Maxwell's Equations Numerical Solvers

This research work analyses techniques for implementing a cell-centred finite-volume time-domain (ccFV-TD) computational methodology for the purpose of studying microwave heating. Various state-of-the-art spatial and temporal discretisation methods employed to solve Maxwell's equations on multidimensional structured grid networks are investigated, and the dispersive and dissipative errors inherent in those techniques examined. Both staggered and unstaggered grid approaches are considered. Upwind schemes using a Riemann solver and intensity vector splitting are studied and evaluated. Staggered and unstaggered Leapfrog and Runge-Kutta time integration methods are analysed in terms of phase and amplitude error to identify which method is the most accurate and efficient for simulating microwave heating processes. The implementation and migration of typical electromagnetic boundary conditions. from staggered in space to cell-centred approaches also is deliberated. In particular, an existing perfectly matched layer absorbing boundary methodology is adapted to formulate a new cell-centred boundary implementation for the ccFV-TD solvers. Finally for microwave heating purposes, a comparison of analytical and numerical results for standard case studies in rectangular waveguides allows the accuracy of the developed methods to be assessed

Direct and inverse problems for diffractive structures - optimization of binary gratings

The goal of the project is to provide flexible analytical and numerical tools for the optimal design of binary and multilevel gratings occurring in many applications in micro-optics. The direct modeling of these diffractive elements has to rely on rigorous grating theory, which is based on Maxwell's equations. We developed efficient and accurate direct solvers using a variational approach together with a generalized finite element method which appears to be well adapted to rather general diffractive structures as well as complex materials. The optimal design problem is solved by minimization algorithms based on gradient descent and the exact calculation of gradients with respect to the geometry parameters of the grating

Divergence preserving discrete surface integral methods for Maxwell's curl equations using non-orthogonal unstructured grids

Several new discrete surface integral (DSI) methods for solving Maxwell's equations in the time-domain are presented. These methods, which allow the use of general nonorthogonal mixed-polyhedral unstructured grids, are direct generalizations of the canonical staggered-grid finite difference method. These methods are conservative in that they locally preserve divergence or charge. Employing mixed polyhedral cells, (hexahedral, tetrahedral, etc.) these methods allow more accurate modeling of non-rectangular structures and objects because the traditional stair-stepped boundary approximations associated with the orthogonal grid based finite difference methods can be avoided. Numerical results demonstrating the accuracy of these new methods are presented

Finite volume approximation of the Maxwell's equations in nonhomogeneous media.

Chung Tsz Shun Eric.Thesis (M.Phil.)--Chinese University of Hong Kong, 2000.Includes bibliographical references (leaves 102-104).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Applications of Maxwell's equations --- p.1Chapter 1.2 --- Introduction to Maxwell's equations --- p.2Chapter 1.3 --- Historical outline of numerical methods --- p.4Chapter 1.4 --- A new approach --- p.5Chapter 2 --- Mathematical Backgrounds --- p.7Chapter 2.1 --- Sobolev spaces --- p.7Chapter 2.2 --- Tools from functional analysis --- p.8Chapter 3 --- Discretization of Vector Fields --- p.10Chapter 3.1 --- Domain triangulation --- p.10Chapter 3.2 --- Mesh dependent norms --- p.11Chapter 3.3 --- Discrete circulation operators --- p.13Chapter 3.4 --- Discrete flux operators --- p.20Chapter 4 --- Spatial Discretization of the Maxwell's Equations --- p.23Chapter 4.1 --- Derivation --- p.23Chapter 4.2 --- Consistency theory --- p.29Chapter 4.3 --- Convergence theory --- p.33Chapter 4.3.1 --- Polyhedral domain --- p.33Chapter 4.3.2 --- Rectangular domain --- p.38Chapter 5 --- Fully Discretization of the Maxwell's Equations --- p.63Chapter 5.1 --- Derivation --- p.63Chapter 5.2 --- Consistency theory --- p.65Chapter 5.3 --- Convergence theory --- p.69Chapter 5.3.1 --- Polyhedral domain --- p.69Chapter 5.3.2 --- Rectangular domain --- p.77Chapter 6 --- Numerical Tests --- p.97Chapter 6.1 --- Convergence test --- p.97Chapter 6.2 --- Electromagnetic scattering --- p.99Bibliography --- p.10