Immersed interface methods have been developed for a variety of
differential equations on domains containing interfaces or irregular
boundaries. The goal is to use a uniform Cartesian grid (or other fixed
grid on simple domain) and to allow other boundaries or interfaces to
cut through this grid. Special finite difference formulas are developed
at grid points near an interface that incorporate the appropriate jump
conditions across the interface so that uniform second-order accuracy
(or higher) can be obtained. For fluid flow problems with an immersed
deformable elastic membrane, the jump conditions result from a balance
between the singular force imposed by the membrane, inertial forces if
the membrane has mass, and the jump in pressure across the membrane.
A second-order accurate method of this type for Stokes flow was developed
with Zhilin Li and more recently extended to the full incompressible
Navier-Stokes equations in work with Long Lee.