We will discuss computational methods for a moving interface in viscous fluid and related error analysis. We will begin with a general description of numerical methods for partial differential equations with discontinuities at interfaces, in which the unknown is calculated at grid points and the (possibly moving) interface is represented separately. We will then describe work with Anita Layton in which we have designed a method for computing the coupled motion of an interface, made of elastic material, with fluid governed by the Navier-Stokes equations. The distinctive aspect of this approach is that we decompose the velocity at each time into a part determined by the Stokes equations (describing viscosity-dominated flow) and the interfacial force, and a ´´regular´´ remainder which can be calculated in a conventional way. Simple test problems indicate that this method is second-order accurate (error O(h^2) where h is the grid spacing), even though the truncation error near the interface is first order. This gain in accuracy has often been observed with methods of this type. For time-independent problems, related to the Laplacian, this observation can be explained by discrete elliptic estimates in maximum norm. For the time-dependent problem, maximum norm estimates for discrete versions of the linear diffusion equation show a regularizing effect, depending on the time discretization, and partially explain the numerical results for fluid flow with interfaces. We have developed versions of these methods which are partially implicit in the interface motion, in order to allow larger time steps. These use a strategy like that in work of Hou, Lowengrub and Shelley.