Diffuse Domain Methods for One and Two-sided Elliptic Interface Problems and Applications

Many important problems in science and engineering involve moving interfaces that separate different materials, fluids, or biological structures. Examples include tumor growth, tissue development, solidification in advanced materials, battery technologies, and complex multiphase fluid flows. Accurately simulating these systems is difficult because the interfaces are often highly irregular, dynamically evolving, and geometrically complex. This project develops new mathematical and computational methods that allow such interface problems to be simulated efficiently and accurately on modern computers without requiring expensive, complicated geometric reconstruction techniques. The resulting tools will support scientific advances in biotechnology, materials science, and advanced manufacturing by enabling faster and more reliable large-scale simulations of complex physical and biological processes. The research will also contribute to the broader scientific computing infrastructure needed for future data-driven and artificial intelligence-assisted modeling technologies. In addition, the project will support the training of graduate students in applied mathematics and scientific computing, produce openly available software for the research community, and expand public access to advanced mathematical education through textbooks and freely available online lecture courses. This project develops and analyzes diffuse domain methods for solving partial differential equations posed on complex and evolving geometries in two and three spatial dimensions. The research focuses on the construction of diffuse approximations of characteristic and surface delta functions associated with curved interfaces and boundaries, together with the derivation of accurate diffuse formulations for one-sided and two-sided interface problems. Analytical tools, including matched asymptotic expansions, weighted Sobolev space techniques, and Gamma-convergence methods, will be used to study consistency, convergence, and stability properties of the resulting approximations as the diffuse interface width tends to zero. The work will also develop fully discrete finite difference and finite element schemes for nonlinear interface problems with nonlinear boundary conditions, emphasizing efficient implementations on regular meshes together with fast multigrid solvers. Freely available software implementations and benchmark datasets will be developed to facilitate the application of these methods to challenging multiphysics problems arising in materials science, biological systems, and complex fluid flows. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria. NSF Award ID: 2607995 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Steven Wise | Institution: University of Tennessee Knoxville, KNOXVILLE, TN | Award Amount: $350,000 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2607995 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2607995.html