| Summary: | The numerical solution of partial differential equations and boundary value problems is one of the most important tools of modern science. For various reasons (parellelizing, improving condition numbers, finding good preconditioners, etc...) it is desirable to turn a boundary value problem over a large domain O into a set of boundary value problems over domains O1,...,O n such that ∪kO k; this is the domain decomposition method. The solutions u1,...,un of the local problems rarely glue together into a solution u of the global problem, hence we must use an iteration whereby we repeatedly solve the local problems. Between each iteration, some information is exchanged between the subdomains, so that the local solutions at the next iteration better approximate the global solution. The method of Schwarz exchanges Dirichlet data along subdomain boundaries, but other methods exist. We recall a construction of nonlocal operators that lead to iterations that converge in 2d + 1 steps, where d is the diameter of the connectivity graph of the domain decomposition, if this graph is a tree. We discuss a graph algorithm linked to these operators in the general case. For the Laplacian on the sphere, we also give local approximations to these optimal nonlocal operators. We also discuss its application for solving the shallow water equations on the sphere as a model for numerical weather prediction.
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