The influence of domain truncation on the performance of optimized Schwarz methods

Abstract

Optimized Schwarz methods enhance convergence using optimized transmission conditions between subdomains. The optimization is usually performed for a model problem on an unbounded domain and two subdomains represented by half spaces. The influence of the domain decomposition geometry on the convergence and the optimized parameters is thus lost in the process, and it is not even theoretically clear if the results published for the unbounded domain still hold in concrete applications where the domains are bounded. We prove here rigorously for a two-subdomain decomposition that the asymptotic performance of optimized Schwarz methods derived from an unbounded domain analysis still holds in the case of a bounded domain, but the constants in the best choice of parameters and convergence rate estimates are influenced by the domain truncation. We obtain accurate estimates for this influence and show theoretically that the domain truncation has more remarkable influence for the slowly converging optimized Schwarz methods than for those converging fast. When the subdomain size is very small, our new optimized parameters lead to much faster algorithms than those obtained from an unbounded domain analysis. We illustrate our theoretical results with numerical experiments.

Synchronous and asynchronous optimized Schwarz methods for one‐way subdivision of bounded domains

“…In the first, we show that if we maintain the height of the domain and the overlap constant, the wider is each subdomain, the bounds on the convergence rates of the synchronous and asynchronous optimized Schwarz methods decrease, indicating a faster converge. This comment is consistent with the observation by Xu 24 for two subdomains (for the synchronous case). Another interpretation of this result is that if we keep the domain fixed, and we increase the number of subdomains, then the convergence factor would deteriorate.…”

“…Step 1. We first prove that ||c 0 (k)|| ∞ < C < ∞, for some C > 0, where c 0 is the initial value of c n as in (24). To that end, we want to bound A 0 s (k) and B 0 s (k), s = 1, … , p, for all k. For ease of notation, for each subdomain s, let the left artificial Dirichlet boundary condition be p s (k) and the right artificial boundary condition q s (k), and let us define = + 2 ∕L 2 2 > 0.…”

Wave-heat coupling in one-dimensional unbounded domains: artificial boundary conditions and an optimized Schwarz method

“…Of course, we considered the simplest setting to start with, and a lot of additional difficulties appear when more complex fluid-structure interaction problems are targeted. For instance, an ongoing work is about the influence of the geometry on the design of optimized conditions, as it has been done in, e.g., [26,46] in another context. As well, two-dimensional and three-dimensional problems should be investigated.…”