Adaptive Multilevel Finite Element Solution of the Poisson-Boltzmann Equation Algorithms I: Algorithms and Examples

This paper is the first of two papers on the adaptive multilevel finite element treatment of the nonlinear Poisson-Boltzmann equation (PBE), a nonlinear elliptic equation arising in biomolecular modeling. Fast and accurate numerical solution of the PBE is usually di#cult to accomplish, due to presence of discontinuous coe#cients, delta functions, three spatial dimensions, unbounded domain, and rapid (exponential) nonlinearity. In this first paper, we explain how adaptive multilevel finite element methods can be used to obtain extremely accurate solutions to the PBE with very modest computational resources, and we present some illustrative examples using two well-known test problems. The PBE is first discretized with piecewise linear finite elements over a very coarse simplex triangulation of the domain. The resulting nonlinear algebraic equations are solved with global inexact-Newton methods, which we have described in a paper appearing previously in this journal [55]. A posteriori error estimates are then computed from this discrete solution, which then drives a simplex subdivision algorithm for performing adaptive mesh refinement. The discretize-solve-estimate-refine procedure is then repeated, until a nearly uniform solution quality is obtained. The sequence of unstructured meshes is used to apply multilevel methods in conjunction with global inexact-Newton methods, so that the cost of solving the nonlinear algebraic equations at each step approaches optimal O(N) linear complexity. All of the numerical procedures are implemented in Manifold Code (MC), a computer program designed and built by the first author over several years at Caltech and UC San Diego. MC is designed solve a very general class of nonlinear elliptic equations on complicated domains in two and thr..