Analysis and Implementation of Isogeometric Boundary Elements for Electromagnetism

Abstract

This thesis is concerned with the analysis and implementation of an isogeometric boundary element method for electromagnetic problems. After an introduction of fundamental notions, we will introduce the electric field integral equation (EFIE), which is a variational problem for the solution of the electric wave equation under the assumption of constant coefficients. Afterwards, we will review the notion of isogeometric analysis, a technique to conduct higher-order simulations efficiently and without the introduction of geometrical errors. We prove quasi-optimal approximation properties for all trace spaces of the de Rham sequence and show inf-sup stability of the isogeometric discretisation of the EFIE. Following the analysis of the theoretical properties, we discuss algorithmic details. This includes not only a scheme for matrix assembly but also a compression technique tailored to the isogeometric EFIE, which yields dense matrices. The algorithmic approach is then verified through a series of numerical experiments concerned with electromagnetic scattering problems. These behave as theoretically predicted. In the last part, the boundary element method is combined with an eigenvalue solver, a so-called contour integral method. We introduce the algorithm and solve electromagnetic resonance problems numerically, where we will observe that the eigenvalue solver benefits from the high orders of convergence offered by the boundary element approach. should be thanked and mentioned for their support. First and foremost, I would like to thank Stefan Kurz and Sebastian Schöps for their extraordinary supervision. The help and care they offer to any of their Ph.D.s are of a quality exceeding any expectations. Moreover, I would like to thank Martin Costabel for the fruitful discussions about my thesis. Many of the results within the thesis have been published already or hopefully will be soon, so I would like to extend my thanks to all of the other coauthors, who helped me along the way, in alphabetical order, namely A. Buffa, J. Dölz, H. Harbrecht, M. Multerer, G. Unger, and R. Vázques. Therein, a special thank you is reserved for Jürgen, whose input was invaluable. Apart from coauthors, I would also like to thank all of my current and former colleagues of CEM, TEMF in general, and the second floor of the GSC, who made my workplace the welcoming and beautiful place it is. Many people gave me their feedback on this manuscript and its early versions, namely Arthur, Hans-Peter, Jürgen, Mehdi, and Shannon, about whose help I am grateful. And finally, I would also like to thank my parents for their help and their support throughout all the years I have spent studying, as well as apologise to anyone who would deserve to be mentioned here but was forgotten, since these few lines have been written at the very end of the entire process and I really want to be done with this. This work is supported by DFG Grants SCHO1562/3-1 and KU1553/4-1, the Excellence Initiative of the German Federal and State Governments and the Graduate School ...