A note onC o Galerkin methods for two-point boundary problems

Summary As is known [4]. theC^o Galerkin solution of a two-point boundary problem using piecewise polynomial functions, hasO(h^2k) convergence at the knots, wherek 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 hasO(h^k+1) convergence, instead ofO(h^k). In this note, it is proved that on any segment there arek–1 interior points where the Galerkin solution is ofO(h^k+2), one order better than the global order of convergence. These points are the Lobatto points.