Existence, Uniqueness, and a Comparison of Nonintrusive Methods for the Stochastic Nonlinear Poisson--Boltzmann Equation

The stochastic nonlinear Poisson-Boltzmann equation describes the electrostatic potential in a ran-dom environment in the presence of free charges and has applications in many fields. We show the existence and uniqueness of the solution of this nonlinear model equation and investigate its regular-ity with respect to a random parameter. Three popular nonintrusive methods, a stochastic Galerkin method, a discrete projection method, and a collocation method, are presented for its numerical so-lution. It is nonintrusive in the sense that solvers and preconditioners for the deterministic equation can be reused as they are. By comparing these methods, it is found that the stochastic Galerkin method and the discrete projection method require comparable computational effort and our results suggest that they outperform the collocation method.