A finite element method scheme for one-dimensional elliptic equations with high superconvergence at the nodes (Q799353)

We set a \(P_ 1\)-type finite element method scheme to approximate one- dimensional elliptic equations. We prove that an appropriate choice of numerical integration formula improves the classical error estimates by a superconvergence result at the nodes of the mesh: \(O(h^ 4)\) instead of \(O(h^ 2)\). This result is actually very cheap: indeed, the integration formulas are easy (Simpson) and, moreover, the structure and the size of the linear system are the same as those of the classical scheme.