We will discuss several projects with the general goal of
designing second-order accurate numerical methods for
the motion of a viscous fluid with a moving interface of
zero thickness which exerts a force in response to its
stretching. The interfacial force results in jumps in
the fluid quantities at the interface. In recent work with
Anita Layton we have found that the problem of the Navier-Stokes
equations with an elastic interface can be simplified by decomposing
the velocity at each time into a ``Stokes'' part, determined
by the (equilibrium) Stokes equations, with the interfacial
force, and a ''regular'' remainder which can be calculated
on a rectangular grid without special treatment at the interface.
For the Stokes part we use the immersed interface method; for the
regular part we use the semi-Lagrangian method. Smaller time
steps can be used to advance the interface with Stokes flow,
using boundary integrals, if needed, to handle the boundary force.
This decomposition exhibits second-order accuracy in
simple test problems. Analytical issues of accuracy and some related
error estimates for the immersed interface method will be
described. We allow for the possibility of more general
boundary motion in work with John Strain for the case of Stokes flow,
in which we use Strain's semi-Lagrangian contouring method
to move the interface. We represent the velocity, on or off
the interface, as a singular integral, and calculate it using
Ewald splitting. The smooth or regularized part is computed as a
Fourier series, while the local part is approximated analytically.