Discontinuous Coarse Spaces for DD-Methods with Discontinuous Iterates

Basic iterative domain decomposition methods (DDM) can only transmit information between direct neighbors. Such methods never converge in less iterations than the diameter of the connectivity graph between subdomains. Convergence rates are dependent on the number of subdomains, and thus algorithms are not scalable. The use of a coarse space [16] is the only way to provide information from distant subdomains, as they enable global information transfer, ensuring scalability. In this respect, well known methods are the two level additive Schwarz method [3], and the FETI [13] and balancing Neumann-Neumann methods [12, 4, 14]. See also [11] for non-symmetric problems. For complete analyses of such scalable methods, see [18, 17]. Adding an effective coarse space correction to an existing method is currently an active area of research, for example in the case of high contrast problems [2, 15]. Combining coarse spaces with methods with discontinuous iterates, such as optimized Schwarz methods (OSM [8]) is also non-trivial, see [6] and chapter 5 in [5] which contain extensive numerical tests, and [7] for a rigorous analysis of a special case. For restricted Additive Schwarz (RAS [1]), which also produces discontinuous global iterates since they are glued from local ones by the R̃ operators in RAS in an aribirary fashion, see [9] in the present proceedings. We explain in §2 why an effective coarse space for non-overlapping OSM (and DDMs with discontinuous iterates in general) should inherently be discontinuous. In §3, we present one possible realization of a coarse grid correction based on a discontinuous coarse space, and we show that convergence in one coarse correction step can be obtained, although this is only practical in one dimension. For higher dimensional problems, we then propose approximations of this optimal coarse space. In §4, we present numerical experiments with this new algorithm, and finally give an outlook on future work in §5.