Moving boundary/interface problems, and problems defined on irregular domains are very challenging both theoretically and numerically. In this talk, I will introduce couple of examples in elasticity and heat transfer to summarize the challenges in the theory and the numerics of these problems. Then I will explain the numerical methods that I have employed to solve those problems.

One of the main components of the numerical methods is the immersed interface method (IIM) developed by myself and LeVeque to solve the governing differential equations involving interfaces and discontinuities. The IIM use Cartesian grids and make use of the given jump conditions. The finite difference schemes are only modified near or on the interface without changing the grid structure. Second order convergence has been proved. Some recent developments include a fast immersed method for Poisson problems with large jumps in the coefficient, the augmented approach for Stokes and Navier-Stokes equations with discontinuous viscosity, source removal technique to avoid computing surface derivatives, and the immersed finite element formulations.

Another major component in solving moving interface problems is how to evolve the interface. In our approach, both the front tracking and the level set methods have been developed for different applications. The level set formulation is used because of the simplicity and robustness for problems involving topological changes and high dimensions. I am going to discuss some issues about how to use the level set method without affecting second order accuracy of the immersed interface method.

Finally, if time allows, some numerical results and simulations will be presented with some physical explanations.

This talk is also an introduction to my (with Dr. Ito) SIAM book: The Immersed Interface Method -- Numerical Solutions of PDEs Involving Interfaces and Irregular Domains.