Local piecewise polynomial projection methods for an O.D.E. which give high-order convergence at knots

Abstract:

Local projection methods which yield ${C^{(m - 1)}}$ piecewise polynomials of order $m + k$ as approximate solutions of a boundary value problem for an mth order ordinary differential equation are determined by the k linear functional at which the residual error in each partition interval is required to vanish on. We develop a condition on these k functionals which implies breakpoint superconvergence (of derivatives of order less than m) for the approximating piecewise polynomials. The same order of superconvergence is associated with eigenvalue problems. A discrete connection between two particular projectors yielding $\mathcal {O}(|\Delta {|^{2k}})$ superconvergence, namely (a) collocation at the k Gauss-Legendre points in each partition interval and (b) "essential least-squares" (i.e., local moment methods), is made by asking that this same order of superconvergence result when using collocation at $k - r$ points per interval and simultaneous local orthogonality of the residual to polynomials of order r; the $k - r$ points then necessarily form a subset of the k Gauss-Legendre points.