On the Schwarz algorithms for the Elliptic Exterior Boundary Value Problems

Abstract

Abstract.Tuning the alternating Schwarz method to the exterior problems is the subject of this paper. We present the original algorithm and we propose a modification of it, so that the solution of the subproblem involving the condition at infinity has an explicit integral representation formulas while the solution of the other subproblem, set in a bounded domain, is approximated by classical variational methods. We investigate many of the advantages of the new Schwarz approach: a geometrical convergence rate, an easy implementation, a substantial economy in computational costs and a satisfactory accuracy in the numerical results as well as their agreement with the theoretical statements.Mathematics Subject Classification. 35J20, 65N38, 65N55.

Iterative methods for scattering problems in isotropic or anisotropic elastic waveguides

“…Let us emphasize that this is possible because the boundary conditions do not couple the modes, contrary to boundary conditions (14) and (27). Besides, let us point out that the iterative processes (35) with boundary conditions (36) or (37) are well-posed, except for a discrete set of frequencies (see Lemma 3.1) that we will recover in the sequel.…”

“…The eigenfrequencies of problems (14)+ (16) and (14)+ (18), associated with algorithm (10)+ (14), are complex when α is complex. However some eigenfrequencies can come close to the real axis.…”

“…A natural idea is to use iterative algorithms instead of direct ones to solve a system involving transparent boundary conditions. This framework has been already applied to several problems set in unbounded domains [14,15,16]. One of the main features is that the system to invert (or equivalently the preconditioner) is chosen as a sparse part of the complete system.…”

Total overlapping Schwarz' preconditioners for elliptic problems

“…We deduce thus that χ = η (in ω γ ) and eventually χ = ϕ (in ω γ ) and η = ϕ (in Ω γ ). A similar approach appeared in [3] and is studied in [4,23] for exterior problems.…”

“…We especially refer to the Fat Boundary Method by Maury (see [6,22,28]), to the Numerical Zoom through the Schwarz Chimera method advocated by Pironneau's team (see [2,7,21,31]) and to the Multi-Scale Multi-Domain Method due the teams of Glowinski and Rappaz (see [14,15]). The modification of the Schwarz method introduced in [4] offers also the opportunity to make local computational zooms when and where needed. Though the 'upgraded' version is with Total Overlap its computational potentialities are substantially enhanced.…”