  For Laplace operator in one space dimension, we propose to formulate the
heuristic finite volume method with the help of mixed Petrov-Galerkin finite
elements. Weighting functions for gradient discretization are parameterized by
some universal function. We propose for this function a compatibility
interpolation condition and we prove that such a condition is equivalent to the
inf-sup property when studying stability of the numerical scheme. In the case
of stable scheme and under two distinct hypotheses concerning the regularity of
the solution, we demonstrate convergence of the finite volume method in
appropriate Hilbert spaces and with optimal order of accuracy.
