  We consider a finite volume scheme with two-point flux approximation (TPFA)
to approximate a Laplace problem when the solution exhibits no more regularity
than belonging to $H^1_0(\Omega)$. We establish in this case some error bounds
for both the solution and the approximation of the gradient component
orthogonal to the mesh faces. This estimate is optimal, in the sense that the
approximation error has the same order as that of the sum of the interpolation
error and a conformity error. A numerical example illustrates the error
estimate in the context of a solution with minimal regularity. This result is
extended to evolution problems discretized via the implicit Euler scheme in an
appendix.
