Approximate solutions to Poisson–Boltzmann systems with Sobolev gradients

A weighted Sobolev gradient approach [1] is presented to a nonlinear PBE [2] with discontinuous coefficient functions. A comparison is given between the weighted and unweighted Sobolev gradient in the finite element setting in two and three dimensions. Behavior of the various Sobolev gradients is discussed for large jump size in the coefficient. A comparison with Newton’s method is given where the failure of Newton’s method is demonstrated for a test problem.