Most biochemical processes involve macromolecules in solution. The
corresponding electrostatics is of central importance for understanding
their structures and functions. An accurate and efficient numerical
scheme is introduced to evaluate the corresponding electrostatic
potential and force by solving the governing Poisson-Boltzmann equation.
This paper focuses on the following issues: (i) the point charge
singularity problem, (ii) the dielectric discontinuity problem across a
molecular surface, and (iii) the infinite domain problem. Green's
function associated with the point charges plus a harmonic function is
introduced as the zeroth order approximation to the solution to solve
the point charge singularity problem. A jump condition capturing finite
difference scheme is adopted to solve the discontinuity problem across
molecule surfaces, where a body-fitting grid is used. The infinite
domain problem is solved by mapping the outer infinite domain into a
finite domain. The corresponding stiffness matrix is symmetric and
positive definite, therefore, fast algorithm such as preconditioned
conjugate gradient method can be applied for inner iteration. Finally,
the resulting scheme is second order accurate for both the potential
and its gradient.