Multigrid methods for Maxwell\u27s equations

In this work we study finite element methods for two-dimensional Maxwell\u27s equations and their solutions by multigrid algorithms. We begin with a brief survey of finite element methods for Maxwell\u27s equations. Then we review the related fundamentals, such as Sobolev spaces, elliptic regularity results, graded meshes, finite element methods for second order problems, and multigrid algorithms. In Chapter 3, we study two types of nonconforming finite element methods on graded meshes for a two-dimensional curl-curl and grad-div problem that appears in electromagnetics. The first method is based on a discretization using weakly continuous P1 vector fields. The second method uses discontinuous P1 vector fields. Optimal convergence rates (up to an arbitrary positive epsilon) in the energy norm and the L2 norm are established for both methods on graded meshes. In Chapter 4, we consider a class of symmetric discontinuous Galerkin methods for a model Poisson problem on graded meshes that share many techniques with the nonconforming methods in Chapter 3. Optimal order error estimates are derived in both the energy norm and the L2 norm. Then we establish the uniform convergence of W-cycle, V-cycle and F-cycle multigrid algorithms for the resulting discrete problems. In Chapter 5, we propose a new numerical approach for two-dimensional Maxwell\u27s equations that is based on the 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. We illustrate this new approach by a P1 finite element method. In Chapter 6, we first report numerical results for multigrid algorithms applied to the discretized curl-curl and grad-div problem using nonconforming finite element methods. Then we present multigrid results for Maxwell\u27s equations based on the approach introduced in Chapter 5. All the theoretical results obtained in this dissertation are confirmed by numerical experiments

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Modeling EMI Resulting from a Signal Via Transition Through Power/Ground Layers

Signal transitioning through layers on vias are very common in multi-layer printed circuit board (PCB) design. For a signal via transitioning through the internal power and ground planes, the return current must switch from one reference plane to another reference plane. The discontinuity of the return current at the via excites the power and ground planes, and results in noise on the power bus that can lead to signal integrity, as well as EMI problems. Numerical methods, such as the finite-difference time-domain (FDTD), Moment of Methods (MoM), and partial element equivalent circuit (PEEC) method, were employed herein to study this problem. The modeled results are supported by measurements. In addition, a common EMI mitigation approach of adding a decoupling capacitor was investigated with the FDTD method

Discontinuous Galerkin Finite Element Methods for Maxwell\u27s Equations in Dispersive and Metamaterials Media

Discontinuous Galerkin Finite Element Method (DG-FEM) has been further developed in this dissertation. We give a complete proof of stability and error estimate for the DG-FEM combined with Runge Kutta which is commonly used in different fields. The proved error estimate matches those numerical results seen in technical papers. Numerical simulations of metamaterials play a very important role in the design of invisibility cloak, and sub-wavelength imaging. We propose a leap-frog discontinuous Galerkin Finite Element Method to solve the time-dependent Maxwell\u27s equations in metamaterials. The stability and error estimate are proved for this scheme. The proposed algorithm is implemented and numerical results supporting the analysis are provided. The wave propagation simulation in the double negative index metamaterials supplemented with perfectly matched layer (PML) boundary is given with one discontinuous Galerkin time difference method (DGTD), of which the stability and error estimate are proved as well in this dissertation. To illustrate the effectiveness of this DGTD, we present some numerical result tables which show the consistent convergence rate and the simulation of PML in metamaterials is tested in this dissertation as well. Also the wave propagation simulation in metamaterals by this DGTD scheme is consistent with those seen in other papers. Several techniques have appeared for solving the time-dependent Maxwell\u27s equations with periodically varying coefficients. For the first time, I apply the discontinuous Galerkin (DG) method to this homogenization problem in dispersive media. For simplicity, my focus is on obtaining a solution in two-dimensions (2D) using 2D corrector equations. my numerical results show the DG method to be both convergent and efficient. Furthermore, the solution is consistent with previous treatments and theoretical expectations

Multi-dimensional Resistivity Models of the Shallow Coal Seams at the Opencast Mine 'Garzweiler I' (Northwest of Cologne) inferred from Radiomagnetotelluric, Transient Electromagnetic and Laboratory Data

The entire Cenozoic unconsolidated fill of the Lower Rhine Embayment in Germany hosts the largest single lignite, or brown coal, deposit in Europe which covers an area of some 2,500 km2 to the northwest of Cologne. Rhineland brown coal is mined in large-scale opencast mining and accounts for around one-quarter of the public electricity supply in Germany. The present study was devoted to carrying out radiomagnetotelluric (RMT) and transient electromagnetic (TEM) investigations over the shallow coal seams at the opencast mine 'Garzweiler I.' The main objectives of the survey were to highlight the applicability and efficiency of RMT and TEM methods in an area like brown coal exploration, and to image the vertical electrical resistivity structure of these coal seams. Therefore, the vertical and lateral resolution capabilities of such methods were as necessary as the ability to cover large areas. Consequently, a total of 86 azimuthal RMT and 33 in-loop TEM soundings were carried out along six separate profiles over two opencast benches at the 'Garzweiler I' mine. The local stratigraphy at the survey areas comprises a layer-cake sequence, from top to bottom, of Garzweiler, Frimmersdorf and Morken coal seams embedded in a sand background, consisting of Surface, Neurath, Frimmersdorf and Morken Sands. A considerable amount of clay and silt intervenes the whole succession. The data were interpreted extensively and consistently in terms of one-dimensional (1D) RMT and TEM resistivity models, without using any complex multi-dimensional interpretation. However, the presence of thin, surficial clay masses (or lenses) broke down such interpretation scheme. In this case, to greatly improve the resistivity resolution for these surficial masses and the underlying coal seams, two-dimensional (2D) RMT and three-dimensional (3D) TEM interpretations have been carried out. They could be used effectively to study the local EM distortion on the measured data, where these surficial masses were found, as well as to cross-check the nearby-topography effect. Because the RMT data are usually skin-depth limited, they only provided a resolution depth between 25 and 30 m for the shallow resistivity structures. Whereas, the TEM data still have sufficiently early- to late-time information, and therefore resulted in a better resolution depth of about 100 m for the shallow to sufficiently-deep resistivity structures. The final 1D/2D RMT and 1D/3D TEM resistivity models displayed a satisfied correlation with both thicknesses derived from the stratigraphic-control boreholes and resistivities measured from direct-current (DC) and spectral induced polarization (SIP) laboratory techniques on 16 rock samples. As demonstrated, the integrated use of azimuthal RMT and in-loop TEM soundings was highly successful and effective at mapping the major stratigraphic units at the survey areas, i.e. the shallowest conductive Garzweiler and Frimmersdorf Coals within their fairly resistive sand background. They could not distinguish between Neurath Sand and the underlying sand/silt or between Frimmersdorf Coal and the underlying organic clay. The deepest Morken Coal was beyond the depth-of-investigation of the present measurements. Finally, the resistivity models revealed that both coal seams gently dip in the southwesterly direction. This should be in fairly good agreement with the regional structural makeup of the Rhineland brown coal. However, they showed that Garzweiler Coal is gradually thinned northeastwards, while Frimmersdorf Coal still has almost a regular thickness

Abstract nonconforming error estimates and application to boundary penalty methods for diffusion equations and time-harmonic Maxwell's equations

International audienceWe devise a novel framework for the error analysis of finite element approximations to low-regularity solutions in nonconforming settings where the discrete trial and test spaces are not subspaces of their exact counterparts}. The key is to use face-to-cell extension operators so as to give a weak meaning to the normal or tangential trace on each mesh face individually for vector fields with minimal regularity and then to prove the consistency of this new formulation by means of some recently-derived mollification operators that commute with the usual derivative operators. We illustrate the technique on Nitsche's boundary penalty method applied to a scalar diffusion equation and to the time-harmonic Maxwell's equations. In both cases, the error estimates are robust in the case of heterogeneous material properties. We also revisit the error analysis framework proposed by Gudi where a trimming operator is introduced to map discrete test functions into conforming test functions. This technique also gives error estimates for minimal regularity solutions, but the constants depend on the material properties through contrast factors

Aeronautical engineering: A continuing bibliography with indexes (supplement 271)

This bibliography lists 666 reports, articles, and other documents introduced into the NASA scientific and technical information system in October, 1991. Subject coverage includes design, construction and testing of aircraft and aircraft engines; aircraft components, equipment and systems; ground support systems; and theoretical and applied aspects of aerodynamics and general fluid dynamics

Queensland University of Technology: Handbook 1995

The Queensland University of Technology handbook gives an outline of the faculties and subject offerings available that were offered by QUT

Queensland University of Technology: Handbook 1995

The Queensland University of Technology handbook gives an outline of the faculties and subject offerings available that were offered by QUT