Generalised Elements on Sheaves and Applications in Spectral Domain Decomposition

This thesis introduces a unified abstract framework for spectral domain decomposition methods, enabling multilevel solvers for a wide range of partial differential equations.  By formalizing the notion of suitable presheaves—generalized function spaces capturing local structure—it defines generalized elements that naturally form a recursive hierarchy. This yields a systematic foundation for multilevel spectral methods, including generalized variants of GenEO and MS-GFEM. The framework encompasses continuous and discontinuous finite element discretizations and provides a rigorous convergence analysis for additive and hybrid Schwarz methods. Numerical experiments for heterogeneous diffusion, linear elasticity, and Helmholtz problems confirm robustness, scalability, and theoretically predicted convergence behavior.