Well-Conditioned Boundary Integral Formulations for Scattering

Description

We consider the scattering of time-harmonic acoustic and electromagnetic waves at bounded objects composed of a few homogeneous linear media (composite scatterer). In this setting wave propagation can be modelled by boundary integral equations, which, after discretization, may serve as the basis for numerical simulation.

A widely used variant, the single trace formulation (STF), relies on unique Dirichlet- and Neumann traces on the union of material interfaces (skeleton). However, the linear systems resulting from its Galerkin boundary element (BEM) discretization will be poorly conditioned whenever fine skeleton meshes are used. For homogeneous scatterers Calderon preconditioning offers a remedy, but its extension to composite scatterers remains unclear.

In this project we aim to investigate completely new boundary integral equations formulations for scattering at composite objects that have been introduced recently. Firstly, the so-called multiple trace boundary integral formulation (MTF) that localizes the traces and reconnects them weakly. There is strong evidence that Calderon preconditioning will perform well for this approach in conjunction with standard Galerkin boundary element discretization. Secondly, a novel 2nd-kind single trace formulation that leads to well-conditioned linear systems without any further treatment.

The numerical analysis of both methods is still incomplete and is to be pursued in the project. In addition, extension to electromagnetic scattering that has not been done yet will be explored. Another open problem is extension to screen-type interfaces. Implementation and extensive numerical tests in two and three dimensions will accompany the theoretical investigations.

Publications / Preprints or SAM-Reports

[1] Claeys, X. and Hiptmair, R. Boundary integral formulation of the first kind for acoustic scattering by composite structures, Submitted, SAM Report 2011/45.
[2] Hiptmair, R. and Jerez-Hanckes, C. Multiple traces boundary integral formulation for helmholtz transmission problems, Submitted, Adv. Appl. Math. (2011), SAM Report 2010/35.
[3] Hiptmair, R. and Mao, S. Stable multilevel splittings of boundary edge element spaces, Submitted, BIT (2011), SAM Report 2011/28.

References

None

Funding

SNF funding applied for.

Contacts

Prof. Ralf Hiptmair