The focus of this work is on the investigation of billiards with its statistical eigenvalue properties. Specifically, superconducting microwave resonators with chaotic characteristics are simulated and the eigenfrequencies that are needed for the statistical analysis are computed. The eigenfrequency analysis requires many (in order of thousands) eigenfrequencies to be calculated and the accurate determination of the eigenfrequencies has a crucial significance. Consequently, the research interests cover all aspects from accurate numerical calculation of many eigenvalues and eigenvectors up to application development in order to get good performance out of the programs for distributed-memory and shared-memory multiprocessors. Furthermore, this thesis provides an overview and detailed evaluation of the used numerical approaches for large-scale eigenvalue calculations with respect to the accuracy, the computational time, and the memory consumption.

The first approach for an accurate eigenfrequency extraction takes into consideration the evaluated electric field computations in Time Domain (TD) of a superconducting resonant structure. Upon excitation of the cavity, the electric field intensity is recorded at different detection probes inside the cavity. Thereafter, Fourier analysis of the recorded signals is performed and by means of signal-processing and fitting techniques, the requested eigenfrequencies are extracted by finding the optimal model parameters in the least squares sense.

The second numerical approach is based on a numerical computation of electromagnetic fields in Frequency Domain (FD) and further employs the Lanczos method for the eigenvalue determination. Namely, when utilizing the Finite Integration Technique (FIT) to solve an electromagnetic problem for a superconducting cavity, which enclosures excited electromagnetic fields, the numerical solution of a standard large-scale eigenvalue problem is considered. Accordingly, if the numerical solution of the same problem is treated by the Finite Element Method (FEM) based on curvilinear tetrahedrons, it yields to the generalized large-scale eigenvalue problem. Afterward, the desired eigenvalues are calculated with the direct solution of the large (generalized) eigenvalue formulations. For this purpose, the implemented Lanczos solvers combine two major ingredients: the Lanczos algorithm with polynomial filtering on the one hand and its parallelization on the other.