We apply the linear embedding via Green's operators (LEGO) method to the scattering by large finite dielectric bodies which contain metallic or penetrable inclusions. After modelling the body by means of LEGO bricks, we formulate the problem via an integral equation for the total incident currents over the boundaries of the bricks. This equation is turned into a weak form by means of the Method of Moments (MoM) and sub-domain basis functions. Then, to handle possibly large MoM matrices, we employ an order-reduction strategy based on: i) compression of the off-diagonal sub-blocks of the system matrix by the adaptive cross approximation algorithm and ii) subsequent compression of the whole matrix by using a basis of orthonormal entire-domain functions generated through the Arnoldi iteration algorithm. The latter leads to a comparatively small upper Hessenberg matrix easily inverted by direct solvers. We validate our approach and discuss the properties of the Arnoldi basis functions through selected numerical examples.
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