### JKU DISSERTATION TNF

A challenging topic in computational electromagnetics is the Maxwell eigenvalue problem. PhD thesis as PDF file 1. 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. This requires the solution of a potential problem, which can be done approximately by a couple of PCG-iterations. Since the desired eigenfunctions belong to the orthogonal complement of the gradient functions, we have to perform an orthogonal projection in each iteration step.

Leszek Demkowicz , University of Texas at Austin. A short outline of the construction is as follows. Since the desired eigenfunctions belong to the orthogonal complement of the gradient functions, we have to perform an orthogonal projection in each iteration step. 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. Further practical advantages will be discussed by means of the following two issues.

Leszek DemkowiczUniversity of Texas at Austin.

## Matus Benko

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

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. This requires diesertation solution of a potential problem, which can be done approximately by a couple of PCG-iterations.

# NUMA – Staff – Benko

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. Since the desired eigenfunctions belong to the orthogonal complement of the gradient functions, we have to perform an orthogonal projection in each iteration step. Further practical advantages will be discussed by means of the following two issues. 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.

The hp -version of FEM combines local mesh refinement h and local increase of the polynomial order of the approximation space p.

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.

## PhD Reviewer Selection and Rigorosum Senate Constitution

For its solution we use the subspace version of the locally optimal preconditioned gradient method. A short outline of the disseertation is as follows. A challenging topic in computational electromagnetics is the Maxwell eigenvalue problem.

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.

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: Numerical examples illustrate dossertation robustness and performance of the method.

In the next step we extend the gradient functions to a hierarchical and conforming basis of the desired polynomial space. 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. An analogous principle is used for the construction of H div -conforming basis functions.