Multilevel methods for the generalized finite element method discretizations

This dissertation is focused on multilevel methods for generalized finite element method (GFEM) discretizations. Our first result is an estimate for the rate of convergence of the method of successive subspace corrections (MSSC) in terms of the method of parallel subspace corrections (MPSC). This is motivated by a new representation for the convergence rate of the MSSC. We also provide some estimates for the convergence rate of multilevel preconditioned systems.;Based on the multilevel preconditioners and the auxiliary space lemma, we study the efficient solution of linear systems arising in discretizations of second order elliptic partial differential equations (PDEs) via the generalized finite element method (GFEM). Our results apply for GFEM equations on unstructured simplicial grids in 2 and 3 spatial dimensions. We then propose a uniform pre-conditioner for GFEM linear systems. Moreover, we show that this result has an application in the design of a multilevel pre-conditioner for the linear elasticity problem.;We have also included our work on applying the multilevel methods to the exact controllability methods for the time-periodic wave equation, which is equivalent to the Helmholtz equation.