2016-04-07

Multi-Trace Boundary Element Methods for Scattering

Prof. Dr. Ralf Hiptmaier, ETH Zurich

9 May 2016, 16:15–17:45; Location: S2|17-103

We consider the scattering of acoustic or electromagnetic waves at a penetrable object composed of different homogeneous materials, that is, the material coefficients are supposed to be piecewise constant in sub-domains. This makes possible to recast the problem into boundary integral equations posed on the interfaces. Those can be discretized by means of boundary elements (BEM). This approach is widely used in numerical simulations and often relies on so-called first-kind single-trace BIE, also known as PMCHWT scheme in electromagnetics. These integral equations directly arise from Calderón identities, but after BEM discretization give rise to poorly conditioned linear systems, for which no preconditioner seems to be available so far.

As a remedy we propose new multi-trace boundary integral equations; whereas the single-trace BIE feature unique Cauchy traces on sub-domain interfaces as unknowns, the multi-trace idea takes the cue from domain decomposition and tears the unknowns apart so that local Cauchy traces are recovered. Two of them live on each interface and thus we dub the methods “multi-trace”. The benefit of localization is the possibility of Calderón preconditioning.

Multi-trace formulations come in two flavors. A first variant, the global multi-trace approach, is obtained from the single-trace equations by taking a “vanishing gap limit” [1]. The second variant is the local multi-trace method and is based on local coupling across sub-domain interfaces [3]. Numerical experiments for acoustic scattering demonstrate the efficacy of Calderón preconditioning.

References

[1] X. Claeys and R. Hiptmair, Multi-trace boundary integral formulation for acoustic scattering by composite structures, Communications on Pure and Applied Mathematics, 66 (2013), pp. 1163–1201.

[2] X. Claeys, R. Hiptmair, C. Jerez-Hanckes, and S. Pintarelli, Novel multi-trace boundary integral equations for transmission boundary value problems, in Unified Transform for Boundary Value Problems: Applications and Advances, A. Fokas and B. Pelloni, eds., SIAM, Philadelphia, 2014, pp. 227–258.

[3] R. Hiptmair and C. Jerez-Hanckes, Multiple traces boundary integral formula-tion for Helmholtz transmission problems, Adv. Comput. Math., 37 (2012), pp. 39–91.

[4] R. Hiptmair, C. Jerez-Hanckes, J.-F. Lee, and Z. Peng, Domain decom-position for boundary integral equations via local multi-trace formulations, in Do-main Decomposition Methods in Science and Engineering XXI., J. Erhel, M. Gander,L. Halpern, G. Pichot, T. Sassi, and O. Widlund, eds., vol. 98 of Lecture Notes inComputational Science and Engineering, Springer, Berlin, 2014, Proceedings of the XXI. International Conference on Domain Decomposition Methods, Rennes, France, June 25-29, 2012 I, pp. 43–58.

Acknowledgement

This talk is based on joint research together with X. Claeys (LJLL, UPMC, Paris) and C. Jerez-Hanckes (Pontificia Universidad Católica de Chile, Santiago de Chile).

Category: CE Seminar

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