Mesh-Centered Finite Differences from Nodal Finite Elements for Elliptic Problems

After it is shown that the classical five-point 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.