The exponentially convergent trapezoidal rule

It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators

The AGE iterative methods for solving large linear systems occurring in differential equations

The work presented in this thesis is wholly concerned with the Alternating Group Explicit (AGE) iterative methods for solving large linear systems occurring in solving Ordinary and Partial Differential Equations (ODEs and PDEs) using finite difference approximations. [Continues.

Simulation and Calibration of the ALICE TPC including innovative Space Charge Calculations

ALICE is one of the four main particle detectors located around the LHC accelerator at CERN. It is particularly designed to study the physics of the quark-gluon plasma by means of nucleus--nucleus collisions at center-of-mass energies up to 5.5 TeV per nucleon pair. A Time-Projection Chamber (TPC) was chosen to be its central-sub-detector due to its low mass properties and its capabilities to provide a robust and accurate Particle Identification even within ultra-high multiplicity environments (up to 8000 tracks per unit of eta). To achieve the required physics performance, the space point resolution of the TPC must be in the order of 0.2 mm. Due to its gigantic size of 5~m in diameter and 5~m in length, corrections for static as well as dynamic effects are indispensable in order to accomplish the design goal. The research presented covers all major issues relevant for the final calibration and therefore the enhancement of the TPC performance in terms of resolution. The main focus was to distinguish between the different effects which disturb the electron trajectory within the drift volume by means of quantifying the magnitude of their influences. The effects were parametrized in terms of physical parameters, as opposed to a multivariate fit, in order to minimize the residuals of the cluster positions. The different chapters of the present research work cover static imperfections, like magnetic and electric field inhom ogeneities due to mechanical imperfections, as well as dynamic variations of the drift properties due to pressure, temperature and gas composition variations which manifest themselves as gas density fluctuations. Furthermore, additional challenges were treated which will occur in future high multiplicity nucleus-nucleus collisions. These are the improvement of the two-track resolution as well as the quantification of additional dynamic field deviations due to space charges. Various simulation techniques were used to qualify and quantify the field imperfections due to mechanical deficiencies. Besides the localization and calibration of the field imperfections, the simulations led to optimized voltage settings which minimize the residuals. The different drift velocity v_d dependencies were parametrized to allow a quick estimation of the dynamic $v_d$ variations as a function of the measured ambient conditions. Besides that, the programmable signal shaping algorithm within the Front-End electronics was revised. This is expected to improve the two-track resolution in high multiplicity events. Moreover, novel analytical solutions were derived to allow a fast and precise prediction of additional dynamic field deviations due to ionic-charge pile up within the TPC gas volume. This analytic approach finally permits accurate simulations of additional systematic shifts along the electron trajectory due to any three dimensional space c harge distribution within the TPC. This innovative method is an essential part of the calibration algorithms which are being developed for the future Pb-Pb collisions at LHC

Small Collaboration: Numerical Analysis of Electromagnetic Problems (hybrid meeting)

The classical theory of electromagnetism describes the interaction of electrically charged particles through electromagnetic forces, which are carried by the electric and magnetic fields. The propagation of the electromagnetic fields can be described by Maxwell's equations. Solving Maxwell's equations numerically is a challenging problem which appears in many different technical applications. Difficulties arise for instance from material interfaces or if the geometrical features are much larger than or much smaller than a typical wavelength. The spatial discretization needs to combine good geometrical flexibility with a relatively high order of accuracy. The aim of this small-scale, week-long interactive mini-workshop jointly organized by the University of Duisburg-Essen and the University of Twente, and kindly hosted at the MFO, is to bring together experts in non-standard and mixed finite elements methods with experts in the field of electromagnetism

International Workshop on Finite Elements for Microwave Engineering

When Courant prepared the text of his 1942 address to the American Mathematical Society for publication, he added a two-page Appendix to illustrate how the variational methods first described by Lord Rayleigh could be put to wider use in potential theory. Choosing piecewise-linear approximants on a set of triangles which he called elements, he dashed off a couple of two-dimensional examples and the finite element method was born. … Finite element activity in electrical engineering began in earnest about 1968-1969. A paper on waveguide analysis was published in Alta Frequenza in early 1969, giving the details of a finite element formulation of the classical hollow waveguide problem. It was followed by a rapid succession of papers on magnetic fields in saturable materials, dielectric loaded waveguides, and other well-known boundary value problems of electromagnetics. … In the decade of the eighties, finite element methods spread quickly. In several technical areas, they assumed a dominant role in field problems. P.P. Silvester, San Miniato (PI), Italy, 1992 Early in the nineties the International Workshop on Finite Elements for Microwave Engineering started. This volume contains the history of the Workshop and the Proceedings of the 13th edition, Florence (Italy), 2016 . The 14th Workshop will be in Cartagena (Colombia), 2018