Multilevel Schwarz methods for Elliptic Partial Differential Equations

We investigate multilevel Schwarz domain decomposition preconditioners, to efficiently solve linear sys- tems arising from numerical discretizations of elliptic partial differential equations by the finite element method. In our analysis we deal with unstructured mesh partitions and with subdomain boundaries resulting from using the mesh partitioner. We start from two-level preconditioners with either aggrega- tive or interpolative coarse level components, then we focus on a strategy to increase the number of lev- els. For all preconditioners, we consider the additive residual update and its multiplicative variants within and between levels. Moreover, we compare the preconditioners behaviour, regarding scalability and rate of convergence. Numerical results are provided for elliptic boundary value problems, including a convection–diffusion problem when suitable stabilization becomes necessary.