  We consider a model convection-diffusion problem and present useful
connections between the finite differences and finite element discretization
methods. We introduce a general upwinding Petrov-Galerkin discretization based
on bubble modification of the test space and connect the method with the
general upwinding approach used in finite difference discretization. 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. This new approach allows for
improving well known upwinding finite difference methods and for obtaining new
error estimates. We prove that the exponential bubble Petrov-Galerkin
discretization can recover the interpolant of the exact solution. As a
consequence, we estimate the closeness of the related finite difference
solutions to the interpolant. The ideas we present in this work, can lead to
building efficient new discretization methods for multidimensional convection
dominated problems.
