Uniform Error Estimates for Navier-stokes Flow with a Moving Boundary Using the Immersed Interface Method

We prove that uniform accuracy of almost second order can be achieved with a finite difference method applied to Navier-Stokes flow at low Reynolds number with a moving boundary, or interface, creating jumps in the velocity gradient and pressure. Difference operators are corrected to O(h) near the interface using the immersed interface method, adding terms related to the jumps, on a regular grid with spacing h and periodic boundary conditions. The force at the interface is assumed known within an error tolerance; errors in the interface location are not taken into account. The error in velocity is shown to be uniformly O(h2| log h|2), even at grid points near the interface, and, up to a constant, the pressure has error O(h2| log h|3). The proof uses estimates for finite difference versions of Poisson and diffusion equations which exhibit a gain in regularity in maximum norm.