  The Immersed Boundary method is a simple, efficient, and robust numerical
scheme for solving PDE in general domains, yet for fluid problems it only
achieves first-order spatial accuracy near embedded boundaries for the velocity
field and fails to converge pointwise for elements of the stress tensor. In a
previous work we introduced the Immersed Boundary Smooth Extension (IBSE)
method, a variation of the IB method that achieves high-order accuracy for
elliptic PDE by smoothly extending the unknown solution of the PDE from a given
smooth domain to a larger computational domain, enabling the use of simple
Cartesian-grid discretizations. In this work, we extend the IBSE method to
allow for the imposition of a divergence constraint, and demonstrate high-order
convergence for the Stokes and incompressible Navier-Stokes equations: up to
third-order pointwise convergence for the velocity field, and second-order
pointwise convergence for all elements of the stress tensor. The method is
flexible to the underlying discretization: we demonstrate solutions produced
using both a Fourier spectral discretization and a standard second-order
finite-difference discretization.
