Adaptive Finite Element Method for Simulation of Optical Nano Structures

We discuss realization, properties and performance of the adaptive finite element approach to the design of nano-photonic components. Central issues are the construction of vectorial finite elements and the embedding of bounded components into the unbounded and possibly heterogeneous exterior. We apply the finite element method to the optimization of the design of a hollow core photonic crystal fiber. Thereby we look at the convergence of the method and discuss automatic and adaptive grid refinement and the performance of higher order elements

Adaptive Local Multigrid Methods for the Solution of Time Harmonic Eddy Current Problems

The efficient computation of large eddy current problems with finite elements requires adaptive methods and fast optimal iterative solvers like multigrid methods. This paper provides an overview of the most important implementation aspects of an adaptive multigrid scheme for time-harmonic eddy currents. It is shown how the standard multigrid scheme can be modified to yield an O(N) complexity even for general adaptive refinement strategies, where the number of unknowns N can grow slowly from one to the next refinement level. Algorithmic details and numerical examples are given

Finite-Element Simulations of Light Propagation through Circular Subwavelength Apertures

Light transmission through circular subwavelength apertures in metallic films with surrounding nanostructures is investigated numerically. Numerical results are obtained with a frequency-domain finite-element method. Convergence of the obtained observables to very low levels of numerical error is demonstrated. Very good agreement to experimental results from the literature is reached, and the utility of the method is demonstrated in the investigation of the influence of geometrical parameters on enhanced transmission through the apertures

Edge Element Approximation for the Spherical Interface Dynamo System

Exploring the origin and properties of magnetic fields is crucial to the development of many fields such as physics, astronomy and meteorology. We focus on the edge element approximation and theoretical analysis of celestial dynamo system with quasi-vacuum boundary conditions. The system not only ensures that the magnetic field on the spherical shell is generated from the dynamo model, but also provides convenience for the application of the edge element method. We demonstrate the existence, uniqueness and stability of the solution to the system by the fixed point theorem. Then, we approximate the system using the edge element method, which is more efficient in dealing with electromagnetic field problems. Moreover, we also discuss the stability of the corresponding discrete scheme. And the convergence is demonstrated by later numerical tests. Finally, we simulate the three-dimensional time evolution of the spherical interface dynamo model, and the characteristics of the simulated magnetic field are consistent with existing work

Method of lines based on finite element technique to analyse electromagnetic problems

A hybrid scheme called finite element method of lines is described and proposed for modelling and analysis of generalized computational electromagnetic problems with emphasis on a number of irregular waveguide. This finite element based method of lines is developed by combining finite element method and the method of lines, so that it not only has high flexibility to treat geometrically and compositionally complex problems but also maintains high accuracy of semi-analytical technique. Analytical and numerical algorithmic building blocks of this new scheme are discussed such as geometry discretization, element mapping, element trial functions, reformulation and computational issues of non-linear ordinary differential equations. The results therefore show that this new technique is able to efficiently solve complex problems as compared with the conventional method of lines. MATLAB was used to compute the solutions of various problems

Numerical Optimization of a Waveguide Transition Using Finite Element Beam Propagation

We develop a general convergence analysis for a class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients