Title: The Immersed Interface Method for Elliptic Partial Differential Equations with Discontinuous Coefficients: Applications to Continuum Solvation and Computational Chemistry

Program: Computing Ph.D. - Computational Math Sciences and Engineering

Advisor: Dr. Oliviero Andreussi

Committee Members: Dr. Donna Calhoun, Dr. Grady Wright 

Abstract

This dissertation develops accurate and efficient numerical methods for solving elliptic partial differential equations with discontinuous coefficients across internal interfaces, with applications to continuum solvation models in computational chemistry. The work comprises three interconnected projects centered on the Immersed Interface Method (IIM), addressing the challenge of maintaining second-order accuracy on Cartesian grids despite sharp coefficient discontinuities at solute-solvent boundaries.

Project 1 establishes the foundational IIM framework for single spherical interfaces representing a molecule in solution. The method constructs correction terms in local surface coordinates, employs radial basis function interpolation on strategically placed control points, and utilizes GMRES iteration coupled with fast Discrete Sine Transform solvers. Validation against analytical solutions demonstrates second-order convergence for both the Dirichlet problem and the generalized Poisson equation with dielectric contrasts representative of aqueous solvation. The solution accuracy proves independent of interface position relative to the Cartesian grid.

Project 2 extends this framework to multi-sphere geometries representing molecular clusters, employing a k-nearest-neighbor (K-NN) RBF strategy and excluding intersecting surface regions. Preliminary results for three intersecting spheres confirm the method's geometric flexibility, though optimization for larger systems remains ongoing.

Project 3 reformulates the entire IIM in Cartesian coordinates, eliminating dependence on surface parameterization, curvature computation, and coordinate transformations. This critical advancement enables practical application to arbitrary molecular surfaces such as proteins, membranes, non-spherical cavities, where surface-aligned approaches become prohibitively complex. Preliminary work demonstrates successful derivation of Cartesian reconstruction coefficients for enforcing flux jump conditions. The single sphere results presented in project 1 are reproduced as well as the multi-sphere case. Validation of results and code optimization is ongoing.