Algebraic two-level preconditioners for the Schur complement method

The solution of elliptic problems is challenging on parallel distributed memory computers since their Green's functions are global. To address this issue, we present a set of preconditioners for the Schur complement domain decomposition method. They implement a global coupling mechanism, through coarse-space components, similar to the one proposed in [Bramble, Pasciak, and Shatz, Math. Comp., 47 (1986), pp. 103-134]. The definition of the coarse-space components is algebraic; they are defined using the mesh partitioning information and simple interpolation operators. These preconditioners are implemented on distributed memory computers without introducing any new global synchronization in the preconditioned conjugate gradient iteration. The numerical and parallel scalability of those preconditioners are illustrated on two-dimensional model examples that have anisotropy and/or discontinuity phenomena.