January 25 - 27, 2007

Combined field integral equations

Ralf Hiptmair (ETH Zürich)

Many frequency domain boundary integral equation (BIE) formulations for exterior boundary value problems from both acoustics (Helmholtz equation) and electromagnetics (vector wave equation) suffer from a notorious instability for wave numbers related to interior resonances. In contrast, the so-called combined field integral equations are not affected.

However, if the boundary is not smooth, the traditional combined field integral equations are no longer L2-coercive, which foils attempts to establish asymptotic quasi-optimality of discrete solutions obtained through conforming Galerkin boundary element schemes.

I am going to survey the derivation and theory of standard combined field integral equations. Then I present new combined field integral equations that possess coercivity in canonical trace spaces even on non-smooth surfaces. The main idea is to use suitable regularizing operators in the framework of both direct and indirect methods. Surface differential operators provide practical choices, which can be implemented with little extra costs. Asymptotic quasi-optimality of conforming boundary elements solutions for the resulting variational problems can be shown.

The regularization idea extends to the case of coupling domain variational formulations and BIE. It can also be applied to Maxwell's equations.

REFERENCES

[1] A. Buffa and R. Hiptmair, A coercive combined field integral equation for electromagnetic scattering, SIAM J. Numer. Anal., 42 (2004), pp. 621-640.

[2] A. Buffa and R. Hiptmair, Regularized combined field integral equations, Numer. Math., 100 (2005), pp. 1-19.

[3] R. Hiptmair and P. Meury, Stabilized fem-bem coupling for helmholtz transmission problems, SIAM J. Numer. Anal., 44 (2006), pp. 2107-2130.

[4] R. Hiptmair and P. Meury, Stabilized FEM-BEM coupling for maxwell transmission problems, tech. rep., SAM, ETH Zürich, 2007. In preparation.

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