Quasi‐local multitrace boundary integral formulations

Abstract

In the context of time harmonic wave scattering by piecewise homogeneous penetrable objects, we present a new variant of the multitrace boundary integral formulation, that differs from the local multitrace approach of (Jerez-Hanckes and Hiptmair, 2012) [13] by the presence of regularization terms involving boundary integral operators and localized at junctions i.e. points where at least three subdomains abut. We prove well-posedness and quasioptimal convergence of conforming Galerkin discretizations for this new formulation, and present numerical results.

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

“…From the previous results, we immediately obtain an estimate on the norm of the projection, which will be key in the analysis of Section 7. Although it shares similarities with Proposition 6.1 of [11], the next result is new. Remark 5.1.…”

“…We deduce in particular that X n (Γ) ⊂ Ker( n • Ψ n ). The proof of the next result is similar to that of Lemma 6.4 from [11].…”

“…Combining the previous two lemmas, we see that ( n • Ψ n )(Id − n • Ψ n ) = 0. From this we deduce immediately the following proposition, which is a variant of Corollary 6.1 from [11].…”

“…We now establish a few results about potential theory that are specific to the multi-domain context. Part of the present section is a variant of results already available in [10,11]. However there are novelties.…”

“…Very similar difficulties arise in a different context: the derivation of Boundary Integral Equations (BIE) adapted to multidomain scattering. The Multi-Trace Formalism (MTF) was introduced in [11,12,14,16] as a complete framework for dealing with multi-domain BIE. From the perspective of functional analysis, MTF offers a clean treatment of cross-points.…”