A two-dimensional model of the hollow cathode and the high-voltage segmented hollow cathode sputtering discharges at high current densities is presented. The main plasma parameters (e.g., complete electron distribution function, spatial distributions of particle densities, gas temperature, etc.) are determined self-consistently. The most important mechanisms contributing to the operating characteristics of these hollow cathode discharges are identified and described quantitatively. The most representative hollow cathode configurations are analyzed and the results are compared to existing experimental data for the helium-argon-copper system, e.g., voltage-current characteristics, density of copper atoms as a function of the discharge current, and percentage argon. Good agreement is found between model predictions and experimental results. The model can be adapted to other high-voltage variants of the hollow cathode discharges.