As is known [4]. the $C^o$ Galerkin solution of a two-point boundary problem using piecewise polynomial functions, has O($h^{2k}$ ) convergence at the knots, where $k$ is the degree of the finite element space. Also, it can be proved [5] that at specific interior points, the Gauss-Legendre points the gradient has O($h^{k+1}$) convergence, instead of O($h^k$). In this note, it is proved that on any segment there are $k–1$ interior points where the Galerkin solution is of O($h^{k+2}$), one order better than the global order of convergence. These points are the Lobatto points.

galerkin methods, two-point boundary problems, lobatto points
Partial differential equations, boundary value problems (msc 65Nxx)
Numerische Mathematik

Bakker, M. (1982). A note on Co Galerkin methods for two-point boundary problems . Numerische Mathematik, 38(3), 447–453.