This paper summarizes the mathematical and numerical theories and computational elements of the adaptive fast multipole Poisson-Boltzmann (AFMPB) solver. We introduce and discuss the following components in order: the Poisson-Boltzmann model, boundary integral equation reformulation, surface mesh generation, the nodepatch discretization approach, Krylov iterative methods, the new version of fast multipole methods (FMMs), and a dynamic prioritization technique for scheduling parallel operations. For each component, we also remark on feasible approaches for further improvements in efficiency, accuracy and applicability of the AFMPB solver to large-scale long-time molecular dynamics simulations. The potential of the solver is demonstrated with preliminary numerical results.
