Application of Galerkin Finite-Element Method with Newton Iterations in Computing Steady-State Solutions of Unipolar Charge Currents in Corona Devices

The phenomenon of corona discharge from thin-wire or sharp-point electrodes has found various important applications especially when effective charging devices are needed. In most parts of the corona charge transport region, the unipolar charge current is established as a consequence of the drifting of ions of single polarity along the electric field lines. With negligible diffusion effects, the equation governing charge transport appears to be of hyperbolic type and has been commonly dealt with by the method of characteristics or with upwind treatments. The Poisson's equation governing the electric potential distribution in the presence of unipolar charge, on the other hand, is of elliptic type and has been conveniently dealt with by the finite-element method or the equivalent for typical boundary-value problems. The first-principle based modeling of the corona device behavior has often involved either back-and-forth iterations of solving for one of the variables with others fixed or using time integrations even when steady states are sought. In the present work, the Galerkin finite-element method is applied uniformly to all the equations in the mathematical system and the Newton iteration method is utilized to obtain quadratically converged steady-state solutions in a few steps. Straightforward application of a typical Galerkin finite-element procedure to all the equations is shown to be quite adequate, because no mechanism for boundary layer formation is present in the unipolar charge current in corona devices. Wiggle-free numerical solutions can be obtained without invoking the upwind schemes or excessive mesh refinements.