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.
Galerkin methods, collocation methods, finite element method, superconvergence, Jacobi polynomials
s.i.a.m.

Bakker, M. (1984). One-dimensional Galerkin methods and superconvergence at interior nodal points.