Keywords: Navier-Stokes flow, Stokes flow, boundary integral, stiff equations, fractional stepping, immersed interface, immersed boundary, semigroups of operators, finite difference methods, parabolic equations, diffusion, regularity, stability, L-stable, A-stable, maximum norm Abstract: We will discuss two related projects. Work with A. Layton has the goal of designing a second-order accurate numerical method for viscous fluid flow with a moving elastic interface with zero thickness, the original problem for which Peskin introduced the immersed boundary method. We will discuss some of the background for such numerical methods. In our approach, we decompose the velocity in the Navier-Stokes equations at each time into a part determined by the (equilibrium) Stokes equations, with the interfacial force, and a "regular" remainder which can be calculated without special treatment at the interface. For the "Stokes" part we use the immersed interface method or boundary integrals; for the regular part we use the semi-Lagrangian method to advance in time. Simple test problems indicate second-order accuracy despite a first-order truncation error near the interface, as has come to be expected with certain interfacial methods. We will describe analytical results which partially justify this expectation. For a fully discrete parabolic equation, we have proved a regularizing effect: If we solve a nonhomogeneous heat equation with a finite difference method, with L-stable temporal discretization, using large time steps, then the solution and its first differences are bounded uniformly by the maximum of the nonhomogeneity, and the second differences are almost bounded. The proof uses the point of view of analytic semigroups of operators.