For brevity, we specialize our discussion to the two dimensional case but its extension to three dimensions follows a similar procedure [1]. We consider M disjoint obstacles each occupying a bounded domain with boundary Γ m for m = 1, . . . M . The unbounded region in the exterior of Γ m is denoted by Ω m . The obstacles are sufficiently separated from each other as to enclose each one with disjoint circular artificial boundaries B m . The computational region Ω - m is bounded internally by the obstacle boundary Γ m and externally by the artificial boundary B m . The unbounded region in the exterior of B m is denoted by Ω + m so that B m is precisely the interface between Ω - m and Ω + m . We also consider the following definitions:
The scattering problem that we are considering consists of the scattering of a plane incident wave, u inc , from multiple soft (Dirichlet) or hard (Neumann) obstacles embedded in the unbounded twodimensional region Ω. As stated in our previous work, the construction of the FEABC is based on a decomposition of the scattered field u into purelyoutgoing wave fields u m , such that u = M m=1 u m in Ω + , where each u m is an outgoing wave radiating from the artificial boundary B m . The fundamental idea of this work is the use of a truncated expansion introduced by Karp in 1961 to represent each
The angular functions F m l (θ m ) and G m l (θ m ) are additional unknowns. They depend on the geometry of the scatterers and the properties of the domains Ω - m . An improved version of the formulation for the scattered field u is given by
for m = 1, . . . M in equations (4)-(6). In (5), ν m denotes the normal derivative on B m .
The symbol H m is the Helmholtz operator in terms of the local polar coordinate system in Ω m . The Eqs. (4)-(5) are the usual continuity of u and its normal derivative at the interface B m . The condition (6) establishes the continuity of the Helmholtz operator at the interface. The system is completed by adding the recurrence formulas for the angular Suggested members of the Scientific Committee: Marcus Grote, Helene Barucq
The weak formulation for this BVP is an extension of the one found in [2] to several obstacles. For the IGA application to the BVP (3)-( 8) with Dirichlet BC, we define the function spaces
for m = 1, . . . M . Then, the weak formulation consists of finding (u,
) ∈ S such that the following equations are satisfied:
The angular function G m l satisfies a similar equation to this last equation. Some of our numerical results obtained by numerically solving (2 )-(8 ) with a second order finite difference approximation in generalized curvilinear coordinates are illustrated by Figs. 1-3. We will present numerical results from the IGA technique at the conference. In the experiments shown above, the numbers of terms, L in the FEABC was increased to achieve the best possible approximation. The overall order of convergence of the combined method for the cylindrical scatterers was two due to the second order of convergence of the numerical method used in the interior. The local nature of the FEABC is of great advantage when compared to alternative ABCs such as Dirichlet to Neumann.