The goal of the presented work is the efficient computation of Maxwell boundary and eigenvalue problems using high order H(curl) finite elements. We discuss a systematic strategy for the realization of arbitrary order hierarchic H(curl)conforming finite elements for triangular and tetrahedral element geometries. The shape functions are classified as lowestorder Nédélec, higher-order edge-based, face-based (only in 3D) and element-based ones. Our new shape functions provide not only the global complete sequence property, but also local complete sequence properties for each edge-, face-, element-block. This local property allows an arbitrary variable choice of the polynomial degree for each edge, face, and element. A second advantage of this construction is that simple block-diagonal preconditioning gets efficient. Our high order shape functions contain gradient shape functions explicitly. In the case of a magnetostatic boundary value problem, the gradient basis functions can be skipped, which reduces the problem size, and improves the condition number. We successfully apply the new high order elements for a 3D magnetostatic boundary value problem, and a Maxwell eigenvalue problem showing severe edge and corner singularities.

This work considers the accurate and efficient finite element simulation of threedimensional eddy current problems. We review the application of H and A based formulations for multiply connected domains for the cases where the conductor has a handle and/or a hole. We focus on an hierarchical hp-finite element discretization of the A based formulation that is gauged by regularization. Based on an explicit kernel splitting of the underlying hp-finite element basis, we present a novel preconditioning technique for eddy current problems. We demonstrate its validity on multiply connected domains and include a series of numerical examples to show the effectiveness of the proposed approach.

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This paper analyzes two-level Schwarz methods for matrices arising from the p-version finite element method on triangular and tetrahedral meshes. The coarse level consists of the lowest order finite element space. On the fine level, we investigate several decompositions with large or small overlap leading to optimal or close to optimal condition numbers. The analysis is confirmed by numerical experiments for a model problem.

Surface acoustic wave filters are widely used for frequency filtering in telecommunications. These devices mainly consist of a piezoelectric substrate with periodically arranged electrodes on the surface. The periodic structure of the electrodes subdivides the frequency domain into stop-bands and pass-bands. This means only piezoelectric waves excited at frequencies belonging to the pass-band-region can pass the devices undamped. The goal of the work presented is the numerical calculation of so-called "dispersion diagrams", the relation between excitation frequency and a complex propagation parameter. The latter describes damping factor and phase shift per electrode. The mathematical model is governed by two main issues, the underlying periodic structure and the indefinite coupled field problem due to piezoelectric material equations. Applying Bloch-Floquet theory for infinite periodic geometries yields a unitcell problem with quasi-periodic boundary conditions. We present two formulations for a frequency-dependent eigenvalue problem describing the dispersion relation. Reducing the unit-cell problem only to unknowns on the periodic boundary results in a small-sized quadratic eigenvalue problem which is solved by QZ-methods. The second method leads to a large-scaled generalized non-Hermitian eigenvalue problem which is solved by Arnoldi methods. The effect of periodic perturbations in the underlying geometry is confirmed by numerical experiments. Moreover, we present simulations of high frequency SAW-filter structures as used in TV-sets and mobile phones.