High-order finite element approximations of the Maxwell equations

Abstract

This thesis discusses numerical approximations of electromagnetic wave propagation, which is mathematically described by the Maxwell equations. These equations are typically either formulated as integral equations or as (partial) differential equations. Throughout this thesis, the numerical discretisation (i.e.~approximation) of the partial differential equations is considered. More specifically, out of the numerous existing discretisation techniques this work focuses on H(curl)-conforming high-order finite element methods (FEM) and high-order discontinuous finite element methods (DG-FEM) for the Maxwell equations. One of the first, and most obvious, questions in designing a high-order FEM and DG-FEM is the choice of basis functions. The Maxwell equations have a special geometric structure. If that is not well-represented by the basis functions, the numerical approximation may lead to spurious, non-physical solutions. Another important feature of a high-order basis is hierarchy. A hierarchic construction makes it easier to use different orders of approximation in different parts of the computational domain. The discussion of a set of basis functions that both preserve the geometric structure of the Maxwell equations -- that is H(curl)-conformity for the formulations used in this thesis -- and have a hierarchic structure forms part of the work presented here.