Abstract or Additional Information
Recently there has been extensive development of numerical methods for fluid flow interacting with moving boundaries or interfaces, using regular finite difference grids which, for simplicity, do not conform to the boundaries. When viscosity is significant it is observed that, with certain methods, the velocity can be accurate to about O(h^2), with grid spacing h, even if the discretization near the interface has truncation error (the error in obeying the equations) as large as O(h). We will describe error estimates which explain how such accuracy can be achieved. We will first discuss maximum norm estimates for finite difference versions of the Poisson equation and diffusion equation with a gain of regularity as for the exact equations. We will then describe the application to the Navier-Stokes equations of fluid flow in a simplified prototype problem. As much as possible we use discrete analogues of standard methods for estimating solutions of partial differential equations.