Numerical solution of the Poisson-Boltzmann equation using tetrahedral finite-element meshes

The automatic three-dimensional mesh generation system for molecular geometries developed in our laboratory is used to solve the Poisson–Boltzmann equation numerically using a finite element method. For a number of different systems, the results are found to be in good agreement with those obtained in finite difference calculations using the DelPhi program as well as with those from boundary element calculations using our triangulated molecular surface. The overall scaling of the method is found to be approximately linear in the number of atoms in the system. The finite element mesh structure can be exploited to compute the gradient of the polarization energy in 10–20% of the time required to solve the equation itself. The resulting timings for the larger systems considered indicate that energies and gradients can be obtained in about half the time required for a finite difference solution to the equation. The development of a multilevel version of the algorithm as well as future applications to structure optimization using molecular mechanics force fields are also discussed. © 1997 John Wiley & Sons, Inc. J Comput Chem 18: 1591–1608, 1997