Adaptive coarse spaces in FETI-DP method for highly heterogeneous problems

This thesis investigates approaches to improve performance of the Finite Element Tearing and Interconnecting Dual Primal (FETI-DP) method in problems with heterogeneously distributed high-contrast coefficients, in which the classical domain decomposition methods struggle. In particular, we focus on adaptive and heuristic approaches for coarse-space enhancement, and we investigate the robustness and stability of the standard strategies based on projections and transformation of basis for incorporating these enhancements. We propose two modifications of existing coarse-space enrichment techniques within FETI-DP, which result in better convergence and improved robustness while maintaining the complexity of the orignal techniques. As the main outcome of the thesis, we introduce two novel techniques: (i) a reduced-basis strategy for the generalized eigenvalue problems appearing in the adaptive approaches and (ii) a heuristic for selecting degrees of freedom to be added to the coarse space. Both techniques allow to identify the ill-posed solution modes while significantly reducing computation cost otherwise pertinent to eigenproblem-based adaptive approaches. The effectiveness of these modifications is illustrated on numerical test problems designed to expose limitations of traditional coarse-space constructions and further tested on systems of equations arising in modular-topology optimization tasks.