A typical elliptic interface problem is cast as piecewise defined elliptic partial differential equations (PDE) in different regions which are coupled with interface conditions, such as jumps in the solution and in the flux across the interface. In many situations, such as when the interface is moving, the challenge is how to solve such a problem accurately, robustly and efficiently without generating a body fitted mesh. The key issue is how to capture the complex geometry of the interface and the jump conditions across the interface effectively on a fixed mesh while the interface is not aligned with the mesh and the PDE is not valid across the interface. The PI proposes a non-traditional finite element method to solve the elliptic and elasticity interface problems. The following are the proposed projects: 1. solving 3D elliptic interface problems; 2. solving 3D elasticity interface problems; 3. solving 3D multi-domain interface problems; 4. solving 2D multi-domain elasticity interface problems; 5. solving moving interface problems; 6. interdisciplinary applications.
Interface problems have applications in electromagnetics, material science, fluid dynamics and other areas. For example, the interface could be the molecular surface of a protein, the water-oil interface, or similar. Therefore, by improving numerical techniques used to model interfaces, the proposed research will have substantial impact on interdisciplinary applications. Further, although the research requires advanced techniques, the concept of an interface exists in everyday life, which means that student interest can be stimulated in applied mathematics seminars.
