Multitrace boundary integral equations

We consider the Helmholtz transmission problem with one penetrable star-shaped Lipschitz obstacle. Under a natural assumption about the ratio of the wavenumbers, we prove bounds on the solution in terms of the data, with these bounds explicit in all parameters. In particular, the (weighted) H 1 norm of the solution is bounded by the L 2 norm of the source term, independently of the wavenumber. These bounds then imply the existence of a resonance-free strip beneath the real axis. The main novelty is that the only comparable results currently in the literature are for smooth, convex obstacles with strictly positive curvature, while here we assume only Lipschitz regularity and star-shapedness with respect to a point. Furthermore, our bounds are obtained using identities first introduced by Morawetz (essentially integration by parts), whereas the existing bounds use the much-more sophisticated technology of microlocal analysis and propagation of singularities. We also adapt existing results to show that if the assumption on the wavenumbers is lifted, then no bound with polynomial dependence on the wavenumber is possible.

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“…• Designing novel integral-equation formulations of the transmission problem; see, e.g., [7] and the reviews [18,19].…”

Section: Introductionmentioning

confidence: 99%

Acoustic transmission problems: Wavenumber-explicit bounds and resonance-free regions

Moiola

Spence

2019

Math. Models Methods Appl. Sci.

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“…A remedy is to resort to spectral or high-order Galerkin discretizations [21,22]. The MTF of the second kind-or global version-can be derived if the fields are sought in terms of suitable linear combinations of layer potentials defined on the union of all interfaces-typically referred to as the skeleton-of material discontinuity [8,18].…”

Section: Introductionmentioning

confidence: 99%

Multitrace/singletrace formulations and Domain Decomposition Methods for the solution of Helmholtz transmission problems for bounded composite scatterers

Jerez-Hanckes

Pérez‐Arancibia

Turc

2017

Journal of Computational Physics

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We present Nyström discretizations of multitrace formulations and non-overlapping Domain Decomposition Methods (DDM) for the solution of Helmholtz transmission problems for bounded composite scatterers with piecewise constant material properties. We investigate the performance of DDM with both classical Robin and generalized Robin boundary conditions. The generalized Robin boundary conditions incorporate square root Fourier multiplier approximations of Dirichlet to Neumann operators. While the classical version of DDM is not particularly well suited for Krylov subspace iterative solvers, we show that the associated DDM linear system can be efficiently solved by hierarchical elimination via Schur complements of the Robin data. We show through numerical examples that the latter version of DDM gives rise to small numbers of Krylov subspace iterations that depend mildly on the frequency and number of subdomains.

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“…For the case n > 1 of (1), the genuine multi-subdomain case, it is mainly first kind BIEs that have been proposed and investigated, see the seminal work [39] (based on [17]) and surveys in [10,9,Section 3 each]. Counterparts for time-harmonic electromagentic scattering, based on the Rumsey principle [42] have been known as PMCHWT BIEs for a long time [5,32,24] and their analysis has been accomplished in [3].…”

Section: Introductionmentioning

confidence: 99%

“…Polynomial Galerkin boundary element methods built on these formulations have to deal with ill-conditioned linear systems on fine meshes [43,Section 4.5] and, as a consequence, with slow convergence of iterative solvers. Preconditioning techniques drawing on ideas from domain decomposition like the Boundary Element Tearing and Interconnecting method (BETI) [35,34,27,29], and Multi-Trace Formulations (MTF) [38,37,8,9,7,25,10] are a remedy, but they entail rather complex algorithms.…”

Section: Introductionmentioning

confidence: 99%

Second kind boundary integral equation for multi-subdomain diffusion problems

2017

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We consider isotropic scalar diffusion boundary value problems whose diffusion coefficients are piecewise constant with respect to a partition of space into Lipschitz subdomains. We allow so-called material junctions where three or more subdomains may abut. We derive a boundary integral equation of the second kind posed on the skeleton of the subdomain partition that involves, as unknown, only one trace function at each point of each interface. We prove the well-posedness of the corresponding boundary integral equations. We also report numerical tests for Galerkin boundary element discretisations, in which the new approach proves to be highly competitive compared to the well-established first kind direct single-trace boundary integral formulation. In particular, GMRES seems to enjoy fast convergence independent of the mesh resolution for the discrete second kind BIE.Keywords second-order transmission problems · boundary integral equations · second kind single integral equations, · boundary element methods Mathematics Subject Classification (2000) 65N12 · 65N38 · 65R20 · 35J05

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Multitrace boundary integral equations

Cited by 28 publications

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Acoustic transmission problems: Wavenumber-explicit bounds and resonance-free regions

Acoustic transmission problems: Wavenumber-explicit bounds and resonance-free regions

Multitrace/singletrace formulations and Domain Decomposition Methods for the solution of Helmholtz transmission problems for bounded composite scatterers

Second kind boundary integral equation for multi-subdomain diffusion problems

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