Multilevel methods such as multigrid and domain decomposition are among the most efficient and scalable solvers developed to date. Adapting them to the next generation of supercomputers and improving their performance and scalability is crucial for ex -ascale computing and beyond. Domain decomposition methods subdivide the global problem into subdomains, and then alternate between local solves and boundary data exchange. This puts significant stress on the network interconnect, since all processes try to communicate at once. On the other hand, during the solve phase, the network is under-utilized. The use of non-blocking communication can only alleviate this issue, but not solve it. In asynchronous methods, on the other hand, computation and communication occur concurrently, with some processes performing computation while others communicate, so that the network is consistently in use.
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