The Poisson - Boltzmann Equation Analysis and Multilevel Numerical Solution

We consider the numerical solution of the Poisson-Boltzmann equation (PBE), a three-dimensional second order nonlinear elliptic partial differential equation arising in biophysics. This problem has several interesting features impacting numerical algorithms, including discontinuous coefficients representing material interfaces, rapid nonlinearities, and three spatial dimensions. Similar equations occur in various applications, including nuclear physics, semiconductor physics, population genetics, astrophysics, and combustion. In this work, we study the PBE, discretizations, and develop multilevel-based methods for approximating the solutions of these types of equations. We first outline the physical model and derive the PBE, which describes the electrostatic potential of a large complex biomolecule lying in a solvent. We next study the theoretical properties of the linearized and nonlinear PBE using standard function space methods; since this equation has not been previously studied theoretically, we provide existence and uniqueness proofs in both the linearized and nonlinear cases. We also analyze box-method discretizations of the PBE, establishing several properties of the discrete equations which are produced. In particular, we show that the discrete nonlinear problem is well-posed. We study and develop linear multilevel methods for interface problems, based on algebraic enforcement of Galerkin or variational conditions, and on coefficient averaging procedures. Using a stencil calculus, we show that in certain simplified cases the two approaches are equivalent, with different averaging procedures corresponding to different prolongation operators. We extend these methods to the nonlinear case through global inexact-Newton iteration, and derive necessary and sufficient descent conditions for the inexact-Newton direction, resulting in extremely efficient yet robust global methods for nonlinear problems. After reviewing some of the classical and modern multilevel convergence theories, we construct a theory for analyzing general products and sums of operators, based on recent ideas from the finite element multilevel and domain decomposition communities. The theory is then used to develop an algebraic Schwarz framework for analyzing our Galerkin-based multilevel methods. Numerical results are presented for several test problems, including a nonlinear PBE calculation of the electrostatic potential of Superoxide Dismutase, an enzyme which has recently been linked to Lou Gehrig’s disease. We present a collection of performance statistics and benchmarks for the linear and nonlinear methods on a number of sequential and parallel computers, and discuss the software developed in the course of the research.