Multigrid Solution of the Poisson-Boltzmann Equation

In order to solve the PB equation with the FAS technique, the charge density of Eq. 40 is inserted as part of the driving term f^k. In addition, a slight modification of the defect correction $\tau^k$ must be added to adjust for the differences of the nonlinear terms on the two scales. Once these changes are made, the solution proceeds just as for the standard elliptic solver, and the same linear scaling is maintained.[42] In the calculations presented here, the total positive and negative charges were maintained constant in the calculation domain. It was found in numerical experiments that the FAS solver obtained the solution with the action converged to a small fraction of k_B T at room temperature within roughly 12 fine scale iterations. Using the same relaxation weighting parameter $\omega$, an SOR solver on the finest scale alone required on the order of 10³ iterations to obtain the same level of convergence in the action, both methods starting from an initial guess of $\phi({\vec r})=0$ everywhere. A cut through the three dimensional potential field is presented in Figure 3. These grid results compare well with an exact numerical solution of the radial PB Eq. for these conditions. The results show that there is promise that the PB fields can be updated quickly enough (roughly 10 lattice sweeps) that the PB solver can be incorporated in simulations of large scale colloid or polyelectrolyte systems.