A meshless, high-order integral equation method for smooth surfaces, with application to biomolecular electrostatics

Abstract

In this thesis, we develop methods for efficient simulation of biomolecular electrostatics based on Poisson-Boltzmann equation. Current techniques using finite-difference solution of differential formulation have many drawbacks. We present an integral formulation that resolves these difficulties and enables an efficient implementation using a recently developed fast solver. The new approach can solve practical engineering problems with good accuracy, but only with an aid of a high quality mesh generator, and sometimes require a large number of panels to discretize a surface. To this end, a novel approach to discretize singular integral equations is proposed. Unlike the traditional boundary element method using panel discretization, the new method is meshless and capable of achieving spectral convergence: numerical errors decrease exponentially fast with increasing size of basis set. We will describe a number of techniques in our approach, including the use of global, high order basis, quadrature-based panel integration, and innovative surface representation. The biomolecular problem is particularly suited for this method because molecular surfaces are typically smooth and can be represented globally using spherical harmonics.

(cont.) The use of flat panels in the traditional approach would incur significant geometrical distortion, in addition to much slower convergence rate. Computational results demonstrate that for a practical problem at engineering accuracy (a tolerance of 10¡3) this new approach requires one to two orders of magnitude fewer unknowns than a flat panel method. For a more stringent tolerance of 10¡6, a comparison to an analytically solvable problem reveals that an improvement more than three orders of magnitude has been achieved.