This paper deals with the electromagnetic characterisation of the transmission and scattering properties of an aperture in a thick conducting plane filled with an inhomogeneous composite material for transverse magnetic (TM) polarisation. Of particular interest in this analysis is the introduction of a new technique that combines the finite element and boundary integral methods. The conjugate gradient method and fast Fourier transform are also incorporated for the solution of the resulting system to allow the treatment of large apertures. Numerical examples are presented which demonstrate the validity, versatility and capability of the technique.