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. 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: A main advantage is that we can choose an arbitrary polynomial order on each edge, face, and cell without destroying the global exact sequence. Since the desired eigenfunctions belong to the orthogonal complement of the gradient functions, we have to perform an orthogonal projection in each iteration step. 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.

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. For its solution we use the subspace version of the locally optimal preconditioned gradient method. A challenging topic in computational electromagnetics is the Maxwell eigenvalue problem. In the next step we extend the gradient functions to a hierarchical and conforming basis of the desired polynomial space. PhD thesis as PDF file 1. 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: A main advantage is that we can choose an arbitrary polynomial order on each edge, face, and cell without destroying the global exact sequence.

## Matus Benko

In the next step we extend the gradient functions to a hierarchical and conforming basis of the desired polynomial space. A challenging topic in computational electromagnetics is the Maxwell eigenvalue problem.

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 key point is to respect the de Rham Complex already in the construction dssertation 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.

Numerical examples illustrate the robustness and performance of the method. This requires the solution of a potential problem, which can be done approximately dissertahion 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.

# PhD Reviewer Selection and Rigorosum Senate Constitution

For its solution we use the subspace version of the locally optimal preconditioned gradient method. The hp -version of FEM hku local mesh refinement h and local increase of the polynomial order of the approximation space p.

Further practical advantages will be discussed by means of the following two issues. The main contribution of this work is a general, unified construction principle for H dissetration – and H dissertation -conforming finite elements of variable and arbitrary order for various element topologies suitable for unstructured hybrid meshes.

The gradient fields of higher-order H 1 -conforming shape functions are H curl -conforming and can be chosen explicitly as shape functions for H curl.

Due to the local exact sequence property this is already satisfied for simple splitting strategies. Since the desired eigenfunctions belong to the orthogonal complement of the gradient functions, we have to perform an orthogonal projection in each iteration step. An analogous principle is used for the construction of H div -conforming basis functions.

Leszek DemkowiczUniversity of Texas at Austin. 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.

dissertaton

# NUMA – Staff – Neumueller

PhD thesis as PDF file 1. 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. A main advantage is that we can choose an arbitrary polynomial order on each edge, face, and cell without destroying the global exact sequence.