  We present a general framework for accurately evaluating finite difference
operators in the presence of known discontinuities across an interface. Using
these techniques, we develop simple-to-implement, second-order accurate methods
for elliptic problems with interfacial discontinuities and for the
incompressible Navier-Stokes equations with singular forces. To do this, we
first establish an expression relating the derivatives being evaluated, the
finite difference stencil, and a compact extrapolation of the jump conditions.
By representing the interface with a level set function, we show that this
extrapolation can be constructed using dimension- and coordinate-independent
normal Taylor expansions with arbitrary order of accuracy. Our method is robust
to non-smooth geometry, permits the use of symmetric positive-definite solvers
for elliptic equations, and also works in 3D with only a change in finite
difference stencil. We rigorously establish the convergence properties of the
method and present extensive numerical results. In particular, we show that our
method is second-order accurate for the incompressible Navier-Stokes equations
with surface tension.
