Immersed Interface Methods for Incompresssible Navier-Stokes Equations Randall J. LeVeque University of Washington Consider an incompressible fluid in which a thin elastic membrane is immersed. The motion of the membrane is then coupled with the motion of the fluid and it is necessary to solve the incompressible Navier-Stokes equations with a singular forcing term along the membrane. This must be coupled with boundary conditions requiring that the elastic and fluid forces balance across membrane, and that the membrane moves with the local fluid velocity. Peskin's immersed boundary method is a popular approach to solving such problems, especially in biofluid dynamics applications. Rather than attempting to follow the deforming geometry with a moving grid, the fluid dynamics equations are solved on a uniform Cartesian grid, with a forcing term given by spreading the singular force to the grid using discrete delta functions of finite width. This method is robust but typically only first order accurate due to this spreading. I will discuss an approach to achieving better accuracy by using the elastic force to impose jump conditions directly on the pressure in the process of solving the elliptic equation that arises in a projection method for the Navier-Stokes equations. Membranes with mass can also be handled, in which case there is an additional inertial force that must be included in the force balance.