The Immersed Interface Method (IIM, LeVeque/Li) was motivated by Peskin's 
Immersed Boundary (IB) Method. The IIM shares many characteristics of the 
IB method. Both methods use simple grid structure. The original motivation 
of the IIM is to improve the accuracy of the IB method from first order to 
second order. This has been achieved by incorporating the jump conditions 
into numerical schemes near or on the interface. In this talk, I will
summarize some recent advances of the IIM, particularly, the applications 
to incompressible Stokes and Navier Stokes equations with singular sources, 
discontinuous viscosity, irregular domains, and free boundary and moving 
interfaces using the augmented IIM. Applications include flow past
cylinders, moving contact line problems, deformable moving interfaces, and
incompressible interfaces in incompressible flows.

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.