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We introduce the fast Encapsulating Domain Decomposition (EDD) scheme for the analysis of radar returns from arbitrarily shaped, elongated open-ended cavities. The scheme is demonstrated here for the 2D-TM case only. The EDD is a hybrid spatial-spectral domain decomposition scheme where the cavity is decomposed into segments along its axis, and the fields on the coupling cross-sections between segments are represented spectrally while those on the cavityÂ¿s walls are represented spatially. This reduces the computational complexity per segment from O(N3) to O(MN2), where N is the number of cross-sectional unknowns and M is the number of wall unknowns per segment, satisfying M<<N, since we construct the segmentation step to be small compared with the cross section.