The molecular electrostatics problem, which asks for the potential generated by a charged solute suspended in a dielectric solvent, is of great importance in computational biology. Poisson models, which treat the solvent as a dielectric continuum, have inherent inaccuracies which can ruin energy predictions. These inaccuracies are primarily due to the inability of continuum models to capture the structure of solvent molecules in close proximity to the solute. A common approach to overcome these inaccuracies is to adjust the dielectric boundary by changing atomic radii. This adjustment procedure can accurately reproduce the expected solvation free energy, but fails to predict thermodynamic behavior. The Solvation Layer Interface Method (SLIC) replaces the standard dielectric boundary condition in Poisson models with a nonlinear boundary condition which accounts for the small-scale arrangement of solvent molecules close to the dielectric interface. Remarkably, SLIC retains the accuracy of Poisson models and furthermore predicts solvation entropies and heat capacities, while removing the need to adjust atomic radii.

In this thesis, we perform foundational numerical analysis for the SLIC model. The first major result is a proof that a solution exists for the nonlinear boundary integral equation arising in the SLIC model. We are able to do this by proving existence for an auxiliary equation whose solutions correspond to the SLIC model's equation. Next, we prove that solutions to the SLIC model are unique for spherical geometries, which are common in biological solutes. Finally, we have experimented with nonlinear solvers for the nonlinear BIE, such as Anderson Acceleration, as well as two discretization techniques, in order to provide scalable numerical methods which can be applied to a variety of problems in drug design and delivery.