An optimized Schwarz method with two‐sided Robin transmission conditions for the Helmholtz equation

Abstract

SUMMARYOptimized Schwarz methods are working like classical Schwarz methods, but they are exchanging physically more valuable information between subdomains and hence have better convergence behavior. The new transmission conditions include also derivative information, not just function values, and optimized Schwarz methods can be used without overlap. In this paper, we present a new optimized Schwarz method without overlap in the 2D case, which uses a different Robin condition for neighboring subdomains at their common interface, and which we call two-sided Robin condition. We optimize the parameters in the Robin conditions and show that for a fixed frequency ω, an asymptotic convergence factor of 1 − O(h 1 4 ) in the mesh parameter h can be achieved. If the frequency is related to the mesh parameter h, h = O( 1 ω γ ) for γ ≥ 1, then the optimized asymptotic convergence factor is 1 − O(ω 1−2γ 8 ). We illustrate our analysis with 2D numerical experiments.

A homographic best approximation problem with application to optimized Schwarz waveform relaxation

“…Similar homographic best approximation problems also occur in the design of optimized Schwarz methods for steady problems, and so far these problems have always been treated by direct analysis; see for example [26,28,27,9] for advection diffusion problems, [7,6,19,16] for indefinite Helmholtz problems, and [12] for the positive definite Helmholtz case. Our results here also apply to homographic best approximation problems from the steady case, and will thus be useful for the further development of optimized Schwarz methods; we currently study the application to indefinite Helmholtz problems.…”

Multigrid methods for Helmholtz problems: A convergent scheme in 1D using standard components

“…The newest variants in this class of preconditioners is called the sweeping preconditioner [18,19]. Also in domain decomposition there are successful preconditioners, with the first fundamental contribution [16], which then led to optimized Schwarz methods for the Helmholtz equation [29,28]. There are also specialized FETI methods, like FETI-H [25], and FETI-DPH [23], with a convergence analysis in [24].…”

Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods

“…We also see that it clearly pays to use optimized parameters, as the iteration count is substantially lower than with the first choice of ik in the transmission conditions. We finally show two numerical experiments from [34] and [32], in order to illustrate that optimized Schwarz methods for Helmholtz equations also work well in more practical situations. In Figure 7, we simulated the approach of an Airbus A340 Fig.…”

“…In addition, it might be possible to choose an even better transmission condition, as indicated toward the end in Lions' work [51], and also by Hagström et al in [42]. All these observations and further developments led at the turn of the century to the invention of the new class of optimized Schwarz methods [33], with specialized variants for Helmholtz problems [34,32]. For an overview for symmetric coercive problems, see [30].…”

“…In addition, to prove convergence for the general overlapping case is an open problem. For the model situation of two subdomains however, one can quantify precisely the dependence of the convergence factor on the wave number k, and the mesh parameter h. We show in Table 5 the resulting convergence factors from [32]. We see that for a fixed wave number k, and a constant overlap, independent of the mesh size h, the algorithm converges with a contraction factor independent of the mesh size h. In the important case where the wave number k scales with the mesh size h like k γ h in order to avoid the pollution effect, see [45,46], we see that the contraction factor only depends very weakly on the growing wave number: for example if the overlap is held constant, all Fourier modes of the error contract at least with a factor 1 − O(k Table 6, we show a numerical experiment for a square cavity open on two sides, and the non-overlapping optimized Schwarz method in order to illustrate the asymptotic results from Table 5.…”