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. 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: Due to the local exact sequence property this is already satisfied for simple splitting strategies. 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 challenging topic in computational electromagnetics is the Maxwell eigenvalue problem. Further practical advantages will be discussed by means of the following two issues.

Leszek Demkowicz , University of Texas at Austin. 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. Since the desired eigenfunctions belong to the orthogonal complement of the gradient functions, we have to perform an orthogonal projection in each iteration step. A challenging topic in computational electromagnetics is the Maxwell eigenvalue problem. A main advantage is that we can choose an arbitrary polynomial order on each edge, face, and cell without destroying the global exact sequence. 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.

A challenging topic in computational electromagnetics is the Maxwell eigenvalue problem. 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 tbf FE-space. Further practical advantages will be discussed by means of the following two issues.

# NUMA – Staff – Benko

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. A short outline of the construction is as follows.

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.

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 mku a correct splitting of the kernel of the curl-operator.

# PhD Reviewer Selection and Rigorosum Senate Constitution

An analogous principle is used for the construction of H div -conforming basis functions. Numerical examples illustrate the robustness and performance of the method. This requires the solution of a potential problem, which can be done approximately by a couple of PCG-iterations.

In the next step we extend the gradient functions to ntf hierarchical and conforming basis of the desired polynomial space. A main advantage is that we can choose an arbitrary polynomial order on each edge, face, and cell without destroying the global exact sequence.

PhD thesis as PDF file 1. For its solution we use the subspace version of the locally optimal preconditioned gradient method. 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: Leszek DemkowiczUniversity of Texas at Austin.

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

Due to the local exact sequence property this is already satisfied for simple splitting strategies. The hp -version of FEM combines local mesh refinement h and local increase of the polynomial order of the approximation space p. 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.