In this paper, electrolyte polymer gels, consisting of a polymer network with ionizable groups and a liquid phase with mobile ions, are investigated. For these gels, we present a volume- and surface-coupled multi-field problem involving chemo-electro-mechanics. First, we derive a convection-diffusion equation for the ion concentrations inside and outside the gel as well as a Laplace equation for the electric field. Second, an equation of motion in order to simulate the unsteady swelling-behavior of the gels, is presented. For the chemo-electro-mechanical coupling, the equations as well as the solution scheme, are given. For the numerical simulation, unconditionally stable, higher order accurate, conservative and implicit space-time finite elements with interpolations - continuous in space and discontinuous in time - are used. We investigate the anionic and the cationic ion concentrations for a given fixed number of bound anionic groups as well as the electric potential inside and outside the gel at a given electric field. The resulting increase in the Donnan potential difference on the anode side of the gel, which represents the higher swelling rate, is in good agreement with experimental results. This shows the validity and the potential of the model.