Robust treatment of cross points in Optimized Schwarz Methods

Abstract

In the field of Domain Decomposition (DD), Optimized Schwarz Method (OSM) appears to be one of the prominent techniques to solve large scale time-harmonic wave propagation problems. It is based on appropriate transmission conditions using carefully designed impedance operators to exchange information between sub-domains. The efficiency of such methods is however hindered by the presence of cross-points, where more than two sub-domains abut, if no appropriate treatment is provided.In this work, we propose a new treatment of the cross-point issue for the Helmholtz equation that remains valid in any geometrical interface configuration. We exploit the multi-trace formalism to define a new exchange operator with suitable continuity and isometry properties. We then develop a complete theoretical framework that generalizes classical OSM to partitions with cross points and contains a rigorous proof of geometric convergence, uniform with respect to the mesh discretization, for appropriate positive impedance operators. Extensive numerical results in 2D and 3D are provided as an illustration of the performance of the proposed method.

Non-self adjoint impedance in Generalized Optimized Schwarz Methods

“…Because this local swapping operator is not continuous when the subdomain partition involves cross-points, in [6] we proposed to replace it by a non-local counterpart which paved the way to a convergence analysis similar to [10] but in general geometrical configurations where cross-points are allowed. This idea was then extended to discrete settings in [8] where an explicit estimate of the convergence rate was provided for general geometrical partitionning, regardless of the presence of cross-points.…”

“…The theorical framework of [6,8] holds for a whole family of impedance operators. It relies on the hypothesis that the impedance is symetric positive definite (SPD), which covers certain OSM approaches that pre-existed in the literature, including the original Després algorithm for which a novel convergence estimate was derived, see Example 11.4 in [8]. However many variants of OSM involve impedance operators that are not SPD and thus cannot be analyzed through the theory of [6,8].…”

Convergence of Restricted Additive Schwarz with impedance transmission conditions for discretised Helmholtz problems

“…The ORAS preconditioner has quite a large literature, e.g., [34,10,25] but, although there are arguments partially explaining its success (e.g., [13], [10, §2.3.2]), there is no rigorous convergence theory for Helmholtz problems. It is worth mentioning that there has been considerable recent interest in convergence theory for non-overlapping domain decomposition methods for Helmholtz problems, e.g., [29,7,9]; these algorithms and the corresponding analyses are quite distinct from the method and analysis given here.…”

Can DtN and GenEO coarse spaces be sufficiently robust for heterogeneous Helmholtz problems?

“…Within the domain decomposition community, there has also been renewed work on additive Schwarz methods, which offer a naturally parallel approach. Following on from the seminal work [18], which utilised Robin (or impedance) transmission conditions to provide a convergent Schwarz method for the Helmholtz problem, a wealth of non-overlapping Schwarz methods have been devised; see the introduction of [13] for a recent overview. In these methods one has to be careful to either avoid or treat cross points (where three or more subdomains meet), as can be done in the robust treatment of [13].…”

“…Following on from the seminal work [18], which utilised Robin (or impedance) transmission conditions to provide a convergent Schwarz method for the Helmholtz problem, a wealth of non-overlapping Schwarz methods have been devised; see the introduction of [13] for a recent overview. In these methods one has to be careful to either avoid or treat cross points (where three or more subdomains meet), as can be done in the robust treatment of [13]. Many optimised approaches rely on deriving higher-order transmission conditions, such as through second order impedance operators in [29], absorbing boundary conditions (ABCs) [10], or non-local operators [15].…”