A multigrid method for the Poisson–Nernst–Planck equations

A computational technique for solving the Poisson–Nernst–Planck (PNP) equations is developed which overcomes the poor convergence rates of commonly used algorithms. The coupled Poisson and charge continuity equations are discretized using an unstructured cell-centered finite volume method. A Newton–Raphson linearization accounting for the coupling between the equations through boundary conditions, and the space charge and drift terms, is developed. The resulting linear system of equations is solved using an algebraic multigrid method, with coarse level systems being created by agglomerating finer-level equations based on the largest coefficients of the Poisson equation. A block Gauss–Seidel update is used as the relaxation method. The method is shown to perform well for the transport of K+K+ and Cl-Cl- in a synthetic ion channel for driving voltages, surface charges, ion concentrations and channel aspect ratios ranging over several orders of magnitude.