This thesis addresses the limitations of the classical condition number-based Conjugate Gradient (CG) iteration bound in solving high-contrast heterogeneous scalar elliptic problems, particularly when employing two-level Schwarz preconditioners. The classical bound, which relies solely on the condition number of the system matrix, fails to accurately predict the convergence behavior in scenarios where the eigenspectrum of the preconditioned system exhibits pronounced clustering and spectral gaps. Motivated by this observation, the thesis develops and analyzes sharpened CG iteration bounds that incorporate detailed spectral information, offering a more nuanced and descriptive understanding of convergence.

Building on foundational work in spectral analysis and iterative solvers, the thesis introduces novel multi-cluster and tail-cluster bounds for the CG method. These bounds are derived through a combination of theoretical analysis and practical algorithms for partitioning eigenspectra, and are validated both analytically and numerically. The new bounds utilize key spectral characteristics, such as cluster condition numbers and spectral width, to more accurately estimate the number of iterations required for convergence. Numerical experiments demonstrate that the sharpened bounds can be up to 1000 times tighter than the classical bound and are effective in distinguishing the robustness of different Schwarz preconditioners.

Despite their improved accuracy, the practical application of these bounds for a priori iteration estimation is challenged by the need for detailed spectral information, which is often unavailable in the early stages of iterative solvers. The thesis discusses heuristic approaches for leveraging partial spectral data and highlights the dependency of bound accuracy on the choice of coefficient functions and preconditioners.

In conclusion, the sharpened CG iteration bounds developed in this work provide a significant advancement in predictive performance analysis for high-contrast elliptic problems. Future research directions include refining cluster partitioning algorithms, improving a priori spectral estimation, and extending the applicability of these bounds to more complex problems and preconditioners.

This thesis contains the development of continuous Kepler orbit- and a discrete numerical integration-based collision detection algorithms in a system of LEO satellites, which in combination with collision algorithm form a simplified space debris evolution model. This model is then used to study the Kessler syndrome. The continuous and discrete algorithms get their names from the solutions of the Two Body Problem (TBP) and the methods for collision detection that they are based on; the analytical and continuous time solution of TBP resulting in the Kepler orbits and the numerical, discrete time Velocity Verlet integration of the TBP. The collision model consists of an algorithm for fragmentation collisions largely based on the NASA Standard Breakup Model and a method for elastic, random scattering collisions. Comparison between the continuous and discrete algorithms shows that on average both predict the same time to the first collision in a system of homogeneously distributed satellites. The algorithms differ in their efficiency depending on the number and the radius of the satellites in and the geometry of the system. For relatively small satellite numbers in large systems, the continuous algorithm is computationally more efficient. However, as more satellites or fragments result from previous collision, the continuous algorithm is outperformed by the discrete algorithm. Consequentially, its time complexity appears to be O(N²). Armed with this knowledge, the continuous algorithm is used to show that an initially small system of satellites is able to evolve into a large population of debris particles within several decades. Similarly, the discrete algorithm is used to show that an ordered collection of satellites in an homogeneously distributed system of debris-like particles exhibits the effect that a collision early on in the simulation can cause a cascade of collisions at a later stage. Hence Both the discrete and continuous algorithms predict a Kessler Syndrome and mimic predictions made by more advanced models from leading space agencies like NASA’s LEGEND, ESA’s DELTA and JAXA’s LEODEEM [Lio+13].Future research could focus on including atmospheric drag and gravitational perturbations to the continuous algorithm, thereby lengthening the time frame during which it can realistically simulate a system of satellites in LEO. To achieve this, it is suggested that one execute the calculations inherent to the algorithm in parallel on a GPU, as these are independent of each other.