High order finite element methods to compute Taylor transfer maps

In beam physics, map methods are important techniques for the design and analysis of lattice structures. The computation of the transfer map for an electric or magnetic element requires the multipole decomposition of the field in the region where the beam passes. In the first part of this dissertation we present new techniques to extract he multipole decomposition of the electric or magnetic field from the measured field data or from the knowledge of the current distribution.; The new high precision technique developed to obtain the multipole decomposition of the field from the measured field data solves the Laplace equation using the Helmholtz vector decomposition theorem and differential algebraic methods. This technique requires the field to be specified on a closed surface enclosing the volume of interest. The method outperforms the conventional finite difference and finite element methods in both the execution speed and the precision achieved. We extend this technique to obtain a verified solution to the Laplace equation by using the Taylor model methods. We also parallelize this technique and implement it on a high performance cluster.; We then present a new high precision technique to find the magnetic field of an arbitrary current distribution. The technique uses the Biot-Savart law and differential algebra methods to compute the magnetic field. Using this technique we develop new computational tools to design accelerator magnets.; Both these techniques can also be combined to solve the Poisson equation when the source distribution is specified inside a volume and the field is specified on the surface enclosing the volume. Besides providing a natural multipole decomposition of the field both these tools have the unique advantage of always producing purely Maxwellian fields.; We demonstrate the utility of these techniques in solving practical problems by applying them to real life applications. We present the design and analysis of a novel combined function multipole magnet with an elliptic cross section that can simplify the correction of aberrations in the large acceptance fragment separators for radioactive ion accelerators. We then apply the Laplace field solver to the measured magnetic field data of the dipole magnet of the MAGNEX spectrometer and extract the multipole decomposition of the magnetic field. Finally, we present the linear and high order ion optic simulations for the proposed design of the superconducting fragment separator (Super-FRS) and also apply the field solver technique to extract the transfer map for the magnetic field data obtained through the TOSCA simulation for the Super-FRS quadrupole magnet.