Preview

PDF, English (Abstract Framework for Spectral Domain Decomposition) - main document Download (14MB) | Lizenz: Generalised Elements on Sheaves and Applications in Spectral Domain Decomposition by Strehlow, Arne Paul underlies the terms of Creative Commons Attribution-NonCommercial-NoDerivatives 4.0

Abstract

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.