  A new scheme is proposed to construct an n-times differentiable function
extension of an n-times differentiable function defined on a smooth domain D in
d-dimensions. The extension scheme relies on an explicit formula consisting of
a linear combination of n+1 function values in D, which extends the function
along directions normal to the boundary. Smoothness tangent to the boundary is
automatic. The performance of the scheme is illustrated by using function
extension as a step in a numerical solver for the inhomogeneous Poisson
equation on multiply connected domains with complex geometry in two and three
dimensions. We show that the modest additional work needed to do function
extension leads to considerably more accurate solutions of the partial
differential equation.
