Non-local variant of the optimised Schwarz method for arbitrary non-overlapping subdomain partitions

Abstract

We consider a scalar wave propagation in harmonic regime modelled by Helmholtz equation with heterogeneous coefficients. Using the Multi-Trace Formalism (MTF), we propose a new variant of the Optimized Schwarz Method (OSM) that remains valid in the presence of cross-points in the subdomain partition. This leads to the derivation of a strongly coercive formulation of our Helmholtz problem posed on the union of all interfaces. The corresponding operator takes the form "identity + non-expansive".

Corners and stable optimized domain decomposition methods for the Helmholtz problem

“…This second test case is devoted to the illustration of the new corner conditions (8). We consider a disk or radius R = 1 inside an hexagon of diameter L = 10, and compute the acoustic field diffracted by an incident plane wave u inc (x, y) = e iωx .…”

“…Therefore, there are still many issues widely open for a rigorous mathematical treatment of optimized DDMs with mesh singularities. In particular, we are not aware of any Partial Differential Equations (PDE) based proof of stability, to which convergence follows, for a DDM with corners, either external ones on the boundary of the domain or internal ones on the interface between two subdomains, except the recent contribution [8] in which a completely different approach is used through the multi-trace formalism.…”

Non-self adjoint impedance in Generalized Optimized Schwarz Methods

“…Until recently, all variants of OSM systematically enforced a coupling between subdomains by means of the same local exchange operator that swapped traces from both sides of each interface. 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.…”

Nonlocal Optimized Schwarz Methods for time-harmonic electromagnetics

“…Several other treatments inspired by available strategies developed for elliptic problems have been proposed for nodal type discretizations [1,22]. Recently the geometric convergence result of [9,10] have been extended to arbitrary geometric partitions, including partitions with cross-points [5,6]. The new approach is based on a novel operator that communicates information globally between subdomains and replaces the standard local exchange operator that operates pointwise on the interface.…”

“…Third, we describe a new treatment of transmission conditions that possibly lead to thick interfaces, see Figure 1. In this new approach, the external boundary of the computational domain is not necessarily part of the skeleton where transmission conditions are imposed, which is new and computationally more optimal compared to [5,6].…”