 
Summary: GENERALIZED COMBINED FIELD INTEGRAL EQUATIONS FOR
THE ITERATIVE SOLUTION OF THE HELMHOLTZ EQUATION IN
THREE DIMENSIONS
XAVIER ANTOINE AND MARION DARBAS
Abstract. This paper addresses the derivation of new secondkind Fredholm combined field
integral equations for the Krylov iterative solution of acoustic scattering problems. These inte
gral equations need the introduction of suitable tangential squareroot operators to regularize the
formulations. Existence and uniqueness occur for these formulations. They can be interpreted as
generalizations of the wellknown BrakhageWerner [A. Brakhage and P. Werner, ¨Uber das Dirich
letsche aussenraumproblem f¨ur die Helmholtzsche schwingungsgleichung, Arch. Math. 16 (1965),
pp. 325329] and Combined Field Integral Equations (CFIE) [R.F. Harrington and J.R. Mautz,
Hfield, Efield and combined field solution for conducting bodies of revolution, Arch. Elektron.
¨Ubertragungstech (AE¨U), 32 (4) (1978), pp. 157164]. Finally, two and threedimensional numeri
cal experiments are performed to test their efficiency.
Key words. Acoustic and electromagnetic scattering, Helmholtz equation, secondkind Fred
holm integral equation, Krylov iterative solution
AMS subject classifications. 35J05, 78A45, 45P05, 47G30, 65F10
1. Introduction. Integral equations are widely used in modern acoustic and
electromagnetic scattering codes for solving large scale problems in the highfrequency
regime [22]. These developments have been strongly influenced by the introduction
