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 highorder finite element methods (FEM) and highorder discontinuous finite element methods (DGFEM) for the Maxwell equations.
One of the first, and most obvious, questions in designing a highorder FEM and DGFEM is the choice of basis functions. The Maxwell equations have a special geometric structure. If that is not wellrepresented by the basis functions, the numerical approximation may lead to spurious, nonphysical solutions. Another important feature of a highorder 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 highorder hierarchic H(curl)conforming FEM is then successfully applied to the timedependent 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 DGFEM 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 firstorder timedependent Maxwell system in one and two dimensions. Having a hierarchic structure of the basis functions is, however, still advantageous in the DGFEM when using varyingorder approximations. In Chapter 4, the H(curl)conforming basis is implemented in the DGFEM framework for the secondorder timeharmonic 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.
Highorder discretisation in space generally requires highorder discretisation in time. One family of the most widelyused highorder timeintegration methods applied in combination with DGFEM are the strongstabilitypreserving RungeKutta (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 fullydiscrete scheme is shown to be computationally more efficient than many of the alternatives using a fixedorder timeintegration 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 timeintegration method is needed that treats the conductivity term in an implicit manner. Chapter 5 describes such timeintegration methods when they are applied to both H(curl)conforming FEM and DGFEM semidiscrete schemes of the secondorder timedependent Maxwell wave equation in threedimensions.
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 SSPRKDG 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 nondissipative. They still induce numerical dispersion but, as in the case of SSPRKDG, this is generally less of a worry if the order of the scheme is increased.
Original language  Undefined 

Awarding Institution 

Supervisors/Advisors 

Thesis sponsors  
Award date  12 Feb 2010 
Place of Publication  Enschede 
Publisher  
Print ISBNs  9789036529686 
DOIs  
Publication status  Published  12 Feb 2010 
Keywords
 METIS270735
 IR69979
 MSC00A72
 Numerical approximations
 EWI17496
 Electromagnetic waves
 Finite element methods
Cite this
Sarmany, D. (2010). Highorder finite element approximations of the Maxwell equations. Enschede: University of Twente. https://doi.org/10.3990/1.9789036529686