On the Scalability of Classical One-Level Domain-Decomposition Methods

On the Scalability of the Schwarz Method

“…Note however that the contraction behaviour for a 1D reaction-diffusion equation is completely different from that of the 1D Laplace equation. This can be proved as shown in [2,3] .…”

“…

Introduction and main resultsAn algorithm is said to be weakly scalable if it can solve progressively larger problems with an increasing number of processors in a fixed amount of time. According to classical Schwarz theory, the parallel Schwarz method (PSM) is not scalable (see, e.g., [2,7]). Recent results in computational chemistry, however, have shed more light on the scalability of the PSM: surprisingly, in contrast with classical Schwarz theory, the authors in [1] provide numerical evidence that in some cases the one-level PSM converges to a given tolerance within the same number of iterations independently of the number N of subdomains.

…”

“…Now, let e 0 : Ω → R be some initialization. The PSM defines the sequences {e n j } n∈N by solving for each n ∈ N the sub-problems (e n j ) ′′ (x) = 0 ∀ x ∈ (a j , b j ), e n j (a j ) = e n−1 j−1 (a j ), e n j (b j ) = e n−1 j+1 (b j ), (2) for each j = 2, . .…”

Methods of Reflections: relations with Schwarz methods and classical stationary iterations, scalability and preconditioning.

“…The first case of scalability (Ω fixed) is widely studied in the literature, see, e.g., [51] and references therein. For the second case (Ω growing) only recently an interesting property was observed for the classical one level Schwarz method in [6], and then theoretically investigated in [8,9,10], for classical one-level Neumann-Neumann, Dirichlet-Neumann and optimized-Schwarz methods see [7]. If a domain decomposition method is not scalable, then a coarse correction can make it scalable.…”

The Domain Decomposition Method of Bank and Jimack as an Optimized Schwarz Method

“…6.1 (Special case of Theorem 2 in[13]). If p 12 = 1 1 lh and p 21 = 1 mh , then the OSM (4.6.1) and (4.6.1) converges independently of the initial guess in 2 iterations, and is thus an optimal Schwarz method.…”