One-dimensional Galerkin methods and superconvergence at interior nodal points

In the case of one-dimensional Galerkin methods the phenomenon of superconvergence at the knots has been known for years [5], [7]. In this paper, a minor kind of superconvergence at specific points inside the segments of the partition is discussed for two classes of Galerkin methods: the Ritz–Galerkin method for $2m$th order self -adjoint boundary problems and the collocation method for arbitrary mth order boundary problems. These interior points are the zeros of the Jacobi polynomial $P_n^{m,m} (\sigma )$ shifted to the segments of the partition; $n = k + 1 - 2m$, where $k$ is the degree of the finite element space. The order of convergence at these points is $k + 2$, one order better than the optimal order of convergence. Also, it can be proved that the derivative of the finite element solution is superconvergent of $O(h^{k + 1} )$ at the zeros of the Jacobi polynomial $P_{n + 1}^{m - 1,m - 1} (\sigma )$ shifted to the segments of the partition. This is one order better than the optimal order of convergence for the derivative.