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The highly versatile finite element-boundary integral method is presented for accurate modeling of fields within cavities recessed in a perfectly conducting, infinite, elliptic cylinder. Properties of flush-mounted cavities and antennas are explored and compared with prior developments. This formulation represents a significant advancement over prior techniques since surface curvature variation, either across a single element or across an array of elements, is now accurately included into the antenna model. An advantage of this approach is the ability to model cavities with curvature varying from planar to the constant curvature of a circular cylinder. Computed eigenvalues for planar-rectangular and cylindrical-rectangular closed cavities are compared with theoretical values. Eigenvalues are presented for elliptic-rectangular cavities. Furthermore, the input impedance of a conformal cavity-backed patch antenna mounted on an elliptic cylinder is presented and compared with a similar antenna on a circular cylinder.