20160407
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 subdomains. 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 socalled firstkind singletrace 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 multitrace boundary integral equations; whereas the singletrace BIE feature unique Cauchy traces on subdomain interfaces as unknowns, the multitrace 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 “multitrace”. The benefit of localization is the possibility of Calderón preconditioning.
Multitrace formulations come in two flavors. A first variant, the global multitrace approach, is obtained from the singletrace equations by taking a “vanishing gap limit” [1]. The second variant is the local multitrace method and is based on local coupling across subdomain interfaces [3]. Numerical experiments for acoustic scattering demonstrate the efficacy of Calderón preconditioning.
References
[1] X. Claeys and R. Hiptmair, Multitrace 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. JerezHanckes, and S. Pintarelli, Novel multitrace 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. JerezHanckes, Multiple traces boundary integral formulation for Helmholtz transmission problems, Adv. Comput. Math., 37 (2012), pp. 39–91.
[4] R. Hiptmair, C. JerezHanckes, J.F. Lee, and Z. Peng, Domain decomposition for boundary integral equations via local multitrace formulations, in Domain 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 2529, 2012 I, pp. 43–58.
Acknowledgement
This talk is based on joint research together with X. Claeys (LJLL, UPMC, Paris) and C. JerezHanckes (Pontificia Universidad Católica de Chile, Santiago de Chile).
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