After it is shown that the classical five points mesh-centered finite difference scheme can be derived from a low order nodal finite element scheme by using nonstandard quadrature formulae, higher order block mesh-centered finite difference schemes for second-order elliptic problems are derived from higher order nodal finite elements with nonstandard quadrature formulae as before, combined to a procedure known as «transverse integration». Numerical experiments with uniform and nonuniform meshes and different types of boundary conditions confirm the theoretical predictions, in discrete as well as continuous norms.