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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 second-kind Fredholm combined field
integral equations for the Krylov iterative solution of acoustic scattering problems. These inte-
gral equations need the introduction of suitable tangential square-root operators to regularize the
formulations. Existence and uniqueness occur for these formulations. They can be interpreted as
generalizations of the well-known Brakhage-Werner [A. Brakhage and P. Werner, ¨Uber das Dirich-
letsche aussenraumproblem f¨ur die Helmholtzsche schwingungsgleichung, Arch. Math. 16 (1965),
pp. 325-329] and Combined Field Integral Equations (CFIE) [R.F. Harrington and J.R. Mautz,
H-field, E-field and combined field solution for conducting bodies of revolution, Arch. Elektron.
¨Ubertragungstech (AE¨U), 32 (4) (1978), pp. 157-164]. Finally, two- and three-dimensional numeri-
cal experiments are performed to test their efficiency.
Key words. Acoustic and electromagnetic scattering, Helmholtz equation, second-kind 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 high-frequency
regime [22]. These developments have been strongly influenced by the introduction
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