Electrical capacitance tomography (ECT) is a non-destructive imaging technique that aims at visualisation of the permittivity distribution of a dielectric object based on the measured capacitance. Volumetric image reconstruction of three-dimensional ECT is a non-linear ill-posed inverse problem. In this paper, the non-linear iteration steps of a regularised Gauss-Newton method have been implemented to solve the inverse permittivity problem of 3D ECT. The forward problem is solved using the finite element method (FEM). The sensitivity matrices are calculated using an adjoined field formulation. The results are presented based on simulated data with additional simulated noise, showing the importance of 3D modelling and non-linear considerations.