High-order finite element approximations of the Maxwell equations

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. In particular, Chapter 3 addresses the major difficulties with the implementation of the basis, especially with ensuring the H(curl)-conforming property of every basis function. In Chapter 5, the high-order hierarchic H(curl)-conforming FEM is then successfully applied to the time-dependent Maxwell wave equation in three spatial dimensions. It is possible to introduce additional flexibility in the FEM by allowing the discrete representation of the solution to be discontinuous. In the resulting DG-FEM it is less important to mimic the geometric structure of the original equations in the definition of the basis functions. Rather, the job is then done by the definition of the numerical fluxes -- the functions that are responsible for coupling the information between the discontinuous elements. As a consequence, the use of nodal basis functions, which are neither hierarchic nor H(curl)-conforming, have proved highly popular, largely because it allows for a very efficient implementation. In Chapter 2, the nodal approach is applied to the first-order time-dependent Maxwell system in one and two dimensions. Having a hierarchic structure of the basis functions is, however, still advantageous in the DG-FEM when using varying-order approximations. In Chapter 4, the H(curl)-conforming basis is implemented in the DG-FEM framework for the second-order time-harmonic Maxwell wave equation in three dimensions. An optimal estimate of the penalty parameter in the numerical flux is derived, which is especially useful in providing guidance on how and where to strike the balance between stability and computational efficiency. High-order discretisation in space generally requires high-order discretisation in time. One family of the most widely-used high-order time-integration methods applied in combination with DG-FEM are the strong-stability-preserving Runge-Kutta (SSPRK) methods. In Chapter 2, the SSPRK method is implemented in a way that its order of accuracy matches that of the space discretisation. The resulting fully-discrete scheme is shown to be computationally more efficient than many of the alternatives using a fixed-order time-integration method. However, if the computational domain contains conductive materials, as is often the case, explicit RK methods, such as SSPRK, are no longer suitable. In that case, a time-integration method is needed that treats the conductivity term in an implicit manner. Chapter 5 describes such time-integration methods when they are applied to both H(curl)-conforming FEM and DG-FEM semi-discrete schemes of the second-order time-dependent Maxwell wave equation in three-dimensions. All the different forms of the Maxwell equations presented in this thesis describe wave propagation. It is therefore natural to ask how and whether a given numerical scheme affects the basic properties of the wave, such as dispersion and dissipation. In Chapter 2, the SSPRK-DG method is shown to be both dispersive and dissipative, albeit the dispersion and dissipation errors are often too small to matter much in many practical applications. By contrast, some of the methods described in Chapter 5 conserve the discrete energy, i.e.~they are non-dissipative. They still induce numerical dispersion but, as in the case of SSPRK-DG, this is generally less of a worry if the order of the scheme is increased.