Adaptive multilevel finite element solution of the Poisson-Boltzmann equation I. Algorithms and examples

This article is the first of two articles On the adaptive multilevel finite element treatment of the nonlinear Poisson-Boltzmann equation (PBE), a non linear eliptic equation arising in biomolecular modeling. Fast and accurate numerical solution of the PBE is usually difficult to accomplish, due to t he presence of discontinuous coefficients, delta functions, three spatial d imensions, unbounded domain, and rapid (exponential) nonlinearity. In this first article, we explain how adaptive multilevel finite element methods ca n be used to obtain extremely accurate solutions to the PBE with very modes t computational resources, and we present some illustrative examples using two well-known test problems. The PBE is first discretized with piece-wise linear finite elements over a very coarse simplex triangulation of the doma in. The resulting nonlinear algebraic equations are solved with global inex act Newton methods, which we have described in an article appearing previou sly in this journal. A posteriori error estimates are then computed from th is 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 obta ined. The sequence of unstructured meshes is used to apply multilevel metho ds in conjunction with global inexact Newton methods, so that the cost of s olving the nonlinear algebraic equations at each step approaches optimal O( N) linear complexity. All of the numerical procedures are implemented. in M ANIFOLD CODE (MC), a computer program designed and built by the first autho r over several years at Caltech and UC San Diego. MC is designed to solve a very general class of nonlinear elliptic equations on complicated domains in two and three dimensions. We describe some of the key features of MC, an d give a detailed analysis of its performance for two model PBE problems, w ith comparisons to the alternative methods. It is shown that the best avail able uniform mesh-based finite difference or box-method algorithms, includi ng multilevel methods, require substantially more time to reach a target PB E solution accuracy than the adaptive multilevel methods in MC. in the seco nd article, we develop an error estimator based on geometric solvent access ibility, and present a series of detailed numerical experiments for several complex biomolecules. (C) 2000 John Wiley & Sons, Inc.