Convergence of Some Two-Level Overlapping Domain Decomposition Preconditioners with Smoothed Aggregation Coarse Spaces

We study two-level overlapping preconditioners with smoothed aggregation coarse spaces for the solution of sparse linear systems arising from finite element discretizations of second order elliptic problems. Smoothed aggregation coarse spaces do not require a coarse triangulation. After aggregation of the fine mesh nodes, a suitable smoothing operator is applied to obtain a family of overlapping subdomains and a set of coarse basis functions. We consider a set of algebraic assumptions on the smoother, that ensure optimal bounds for the condition number of the resulting preconditioned system. These assumptions only involve geometrical quantities associated to the sub domains, namely the diameter of the sub domains and the overlap. We first prove an upper bound for the condition number, which depends quadratically on the relative overlap. If additional assumptions on the coarse basis functions hold, a linear bound can be found. Finally, the performance of the preconditioners obtained by different smoothing procedures is illustrated by numerical experiments for linear finite elements in two dimensions.