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This paper deals with the upwind finite volume method applied to the linear advection equation on a bounded domain and with natural boundary conditions. We introduce what we call the geometric corrector, which is a sequence associated with every finite volume mesh in $\mathbf{R}^{nd}$ and every nonvanishing vector $\mathbf{a}$ of $\mathbf{R}^{nd}$. First we show that if the continuous solution is regular enough and if the norm of this corrector is bounded by the mesh size, then an order one error estimate for the finite volume scheme occurs. Afterwards we prove that this norm is indeed bounded by the mesh size in several cases, including the one where an arbitrary coarse conformal triangular mesh is uniformly refined in two dimensions. Computing numerically exactly this corrector allows us to state that this result might be extended under conditions to more general cases, such as the one with independent refined meshes. [ABSTRACT FROM AUTHOR]