  We present useful connections between the finite difference and the finite
element methods for a model boundary value problem. We start from the
observation that, in the finite element context, the interpolant of the
solution in one dimension coincides with the finite element approximation of
the solution. This result can be viewed as an extension of the Green function
formula for the solution at the continuous level. We write the finite
difference and the finite element systems such that the two corresponding
linear systems have the same stiffness matrices and compare the right hand side
load vectors for the two methods. Using evaluation of the Green function, a
formula for the inverse of the stiffness matrix is extended to the case of
non-uniformly distributed mesh points. We provide an error analysis based on
the connection between the two methods, and estimate the energy norm of the
difference of the two solutions. Interesting extensions to the 2D case are
provided.
