This thesis deals with the higher-order Finite Element Method (FEM) for computational electromagnetics.
The hp-version of FEM combines local mesh refinement (h) and local increase of the
polynomial order of the approximation space (p).
A key tool in the design and the analysis of numerical methods for electromagnetic problems is the de
Rham Complex relating the function spaces H1(Ω), H(curl, Ω),
H(div, Ω), and L2(Ω) and their natural differential operators.
For instance, the range of the gradient operator on H1(Ω) is spanned by the space
of irrotional vector fields in H(curl), and the range of the curl-operator on
H(curl, Ω) is spanned by the solenoidal vector fields in H(div, Ω).
The main contribution of this work is a general, unified construction principle for
H(curl)- and H(div)-conforming finite elements of variable and arbitrary order for
various element topologies suitable for unstructured hybrid meshes. The key point is to respect
the de Rham Complex already in the construction of the finite element basis functions and not,
as usual, only for the definition of the local FE-space. A short outline of the construction is as
follows. The gradient fields of higher-order H1-conforming shape functions are
H(curl)-conforming and can be chosen explicitly as shape functions for H(curl).
In the next step we extend the gradient functions to a hierarchical and conforming basis of the desired
polynomial space. An analogous principle is used for the construction of H(div)-conforming basis
functions. By our separate treatment of edge-based, face-based, and cell-based functions, and by including
the corresponding gradient functions, we can establish the local exact sequence property: the subspaces
corresponding to a single edge, a single face or a single cell already form an exact sequence.
A main advantage is that we can choose an arbitrary polynomial order on each edge, face, and cell
without destroying the global exact sequence.
Further practical advantages will be discussed by means of the following two issues.
The main difficulty in the construction of efficient and parameter-robust preconditioners for
electromagnetic problems is indicated by the different scaling of solenoidal and irrotational fields
in the curl-curl problem. Robust Schwarz-type methods for Maxwell's equations rely on a FE-space
splitting, which also has to provide a correct splitting of the kernel of the curl-operator.
Due to the local exact sequence property this is already satisfied for simple splitting strategies.
Numerical examples illustrate the robustness and performance of the method.
A challenging topic in computational electromagnetics is the Maxwell eigenvalue problem.
For its solution we use the subspace version of the locally optimal preconditioned gradient method.
Since the desired eigenfunctions belong to the orthogonal complement of the gradient functions,
we have to perform an orthogonal projection in each iteration step. This requires the solution
of a potential problem, which can be done approximately by a couple of PCG-iterations. Considering
benchmark problems involving highly singular eigensolutions, we demonstrate the performance of the
constructed preconditioners and the eigenvalue solver in combination with hp-discretization on
geometrically refined, anisotropic meshes.