Domain Decomposition Based Hybrid Methods of Finite Element and Finite Difference and Applications in Biomolecule Simulations

Abstract

The dielectric continuum models, such as Poisson Boltzmann equation (PBE), size modified PBE (SMPBE), and nonlocal modified PBE (NMPBE), are important models in predicting the electrostatics of a biomolecule in an ionic solvent. To solve these dielectric continuum models efficiently, in this dissertation, new finite element and finite difference hybrid methods are constructed by Schwartz domain decomposition techniques based on a special seven-box partition of a cubic domain. As one important part of these methods, a finite difference optimal solver --- the preconditioned conjugate gradient method using a multigrid V-cycle preconditioner --- is described in details and proved to have a convergence rate independent of mesh size in solving a symmetric positive definite linear system. These new hybrid algorithms are programmed in Fortran, C, and Python based on the efficient finite element library DOLFIN from the FEniCS project, and are well validated by test models with known analytical solutions. Comparison numerical tests between the new hybrid solvers and the corresponding finite element solvers are done to show the improvement in efficiency. Finally, as applications, solvation free energy and binding free energy calculations are done and then compared to the experiment data.