The influence of point contacts as constituents of imperfect contacts on the total impedance is treated rigorously in two dimensions by finite element simulations. The frequency dependence of the electrical potential distribution and the impedance spectra are calculated for different electrode geometries. It is shown how the existence of insulating gaps at the electrode/solid electrolyte interface results in two semicircles in the complex impedance plane. The elements of the corresponding equivalent circuit are, interpreted in terms of geometrical parameters and materials properties. A model is presented which allows one to understand this behaviour and to predict the influence of conductivity and dielectric constants on the impedance spectra. The results are also applicable to imperfect contacts in bicrystals and polycrystalline materials.