Multilevel optimized schwarz methods

We define a new two-level optimized Schwarz method (OSM), and we provide a convergence analysis both for overlapping and nonoverlapping decompositions. The two-level analysis suggests how to choose the optimized parameters. We also discuss an optimization procedure which relies only on the already studied one-level min-max problems, and we show that these two approaches are asymptotically equivalent. The two-level OSM has mesh independent convergence and it is scalable. We then generalize the two-level method defining a multilevel domain decomposition method which uses the OSM as a smoother. The main advantage of the method consists of its robustness and generality with respect to the equations under study. Thanks to the smoothing properties of the OSM, both with and without overlap, we can define a unique algorithm which can be applied to several equations, both with homogeneous and heterogeneous coefficients. We present extensive numerical results to compare the multilevel OSM, the one-level OSM, and the multigrid scheme. The experiments show that the multilevel OSM inherits robustness from the one-level OSM for heterogeneous elliptic problems, wave problems, and heterogeneous couplings. Finally, we apply the method to design a two-level solver for the heterogeneous Stokes-Darcy system.