A new box iterative method for a class of nonlinear interface problems with application in solving Poisson–Boltzmann equation

• Present a new box iterative method for a class of nonlinear interface problems.
• Obtain a new hybrid solver of Poisson–Boltzmann Equation (PBE) as application.
• Develop a new Newton-PCG-MG scheme for nonlinear boundary value problems.
• Obtain a simple series solution for Poisson ball test model with multiple charges.
• Validate the new PBE hybrid software and demonstrate its high performance.

In this paper, a new box iterative method for solving a class of nonlinear interface problems is proposed by intermixing linear and nonlinear boundary value problems based on a special seven-overlapped-boxes partition. It is then applied to the construction of a new finite element and finite difference hybrid scheme for solving the Poisson–Boltzmann equation (PBE) — a second order nonlinear elliptic interface problem for computing electrostatics of an ionic solvated protein. Furthermore, a modified Newton minimization algorithm accelerated by a multigrid preconditioned conjugate gradient method is presented to efficiently solve each involved nonlinear boundary value problem. In addition, the analytical solution of a Poisson dielectric test model with a spherical solute region containing multiple charges is expressed in a simple series of Legendre polynomials, resulting in a new PBE test model that works for a large number of point charges. The new PBE hybrid solver is programmed as a software package, and numerically validated on the new PBE test model with 892 point charges. It is also compared to a commonly used finite difference scheme in the accuracy of computing solution and electrostatic free energy for three proteins with up to 2124 atomic charges. Numerical results on six proteins demonstrate its high performance in comparison to the PBE finite element program package reported in Xie (2014).