Asynchronous methods refer to parallel iterative procedures where each process performs its task without waiting for other processes to be completed, i.e., with whatever information it has locally available and with no synchronizations with other processes. For the numerical solution of a general partial differential equaition on a domain, Schwarz iterative methods use a decomposition of the domain into two or more (possibly overlapping) subdomains. In essence one is introducing new artificial boundary conditions on the interfaces between these subdomains. In the classical formulation, these artificial boundary conditions are of Dirichlet type. Given an initial approximation, the method progresses by solving for the PDE restricted to each subdomain using as boundary data on the artificial interfaces the values of the solution on the neighboring subdomain from the previous step. This procedure is inherently parallel, since the (approximate) solutions on each subdomain can be performed by a different processor. In the case of optimized Schwarz, the boundary conditions on the artificial interfaces are of Robin or mixed type. In this way one can optimize the Robin parameter(s) and obtain a very fast method.

In this talk, an asynchronous version of the optimized Schwarz method is presented for the solution of differential equations on a parallel computational environment. In a one-way subdivision of the computational domain, with overlap, the method is shown to converge when the optimal artificial interface conditions are used. Convergence is also proved under very mild conditions on the size of the subdomains, when approximate (non-optimal) interface conditions are utilized. Numerical results are presented on large three-dimensional problems illustrating the efficiency of the proposed asynchronous parallel implementation of the method.

This is joint work with Frederic Magoues and Cedric Venet.