Abstract The paper investigates the truncation error between the Green function and the lattice Green function (LGF) for the Laplacian operator defined on the $2$-torus and its discretization on a regular square lattice. Extensions to the cylinder and the rectangular domain with free (or Neumann) boundary conditions are also proposed. In each of these instances, the Green function and its discrete analog are given in exact analytical closed-form allowing to infer accurate estimates as the lattice spacing tends to zero. As expected, it is shown that the continuum limit of the LGF coincides well with the Green function in every case. In particular, the issue of logarithmic singularity regularization of the Green function by the lattice discretization is addressed through two related application examples regarding the rectangular domain, and devoted to the computation of corner-to-corner resistance of an electrical conducting square and the mean first-passage time between the diagonally opposite vertices of a square for a standard Brownian motion, both derived considering the continuum limit.