This lecture is based on a joint on-going project with F. Cakoni E. Darrigrand. It is concerned with a modified Despr´es integral equation method [1], which can be applied to the transmission problem for the Helmholtz equations with nonhomogeneous mediums. By using appropriate relations between the Cauchy data on the interface for the Helmholtz equation in the interior of the scatterer, the transmission conditions can be rewritten in the form an impedance condition for the exterior problem. This impedance condition can be shown to satisfy a necessary condition in order to apply the Despr¨es integral equation method. By substituting the derived impedance condition into the Calderon projector for the Helmholtz equation in the exterior domain, we then obtain a resulting system of integral equations which consists of only boundary integral equations of the second kind and is suitable for using iterative schemes for numerical approximations. As in the previous work by Darrigrand [2], one may couple the fast multiple and the micro-local discretization methods for treating the resulting system. The formal will accelerate the iteration solution of the system and the latter can be used to deal with transmission problem with high frequencies.
References
[1] B. Despr´es, Fonctionnelle quadratique et ´equations pour les probl´emes
d’onde harmonique en domaine ext´erieur, Mod. Math. Anal. Numer. 31(6),
679 (1997).
[2] E. Darrigrand, Coupling of fast multipole method and microlocal discretization
for the 3-D Helmholtz equation, J. Comput. Phys. 181, 126–154
(2002).