Computer calculations have been made for a plasma composed of electron and ion fluids, with the electrostatic field as an additional variable described by Poisson's equation, allowing for the first time a hydrodynamic study of the electrostatic phenomena in plasmas, where very short time steps, large computer capacity and special numerical procedures were used. The numerically observed oscillations of the fields and electron fluid and the waves and their damping by collisions have been evaluated and an analytical model has been derived to study damping by Coulomb collisions. The magnitudes of the electrostatic fields in inhomogeneous plasma have been evaluated as functions of temperature and density gradient. For laser irradiation of 1016 W/cm2, including nonlinear forces with the complete intensity dependent (nonlinear) optical response for heating and dielectric force effects, electrostatic fields exceeding 108 V/cm are generated.