The excitation of harmonics in the current of capacitive radio frequency (RF) discharges is a frequently observed phenomenon. The effect is of interest for several reasons. It forms, for instance, the basis of a successful diagnostic concept for technical plasmas, and it is intimately connected to the process of electron heating in capacitive discharges. Recently, mathematical models were proposed which interpret the phenomenon as the self-excitation of the plasma series resonance by the nonlinearity of the boundary sheath. These models are surprisingly successful but suffer from the limitation that they analyse the plasma dynamics in terms of global equations with concentrated parameters. They are unable to account for the complex multi-mode current waveforms seen in experiments and also cannot resolve the electromagnetic effects which dominate contemporary/large area, high frequency processing discharges. This paper aims to correct the deficiency by presenting a spatially resolved, fully electromagnetic model of the nonlinear RF dynamics of a bounded plasma. The model holds for arbitrary plasma reactor geometries and external RF excitations and makes no assumptions on the homogeneity of the plasma or the characteristics of the boundary sheath. A functional analytic (Hilbert space) formulation of the model is given which allows for an exact solution in terms of an infinite power/Fourier series. For the case of an idealized cylindrical reactor, the model is also explicitly evaluated. The calculated RF current wave forms exhibit the complex multi-mode structure of the currents observed in experiments and follow the same scaling laws. It is concluded that the presented model is capable of describing the nonlinear dynamics of capacitive RF discharges of all sizes and that it provides a significant improvement over both the established nonlinear global models and linear models with spatial resolution.