Discrete petrov-galerkin scheme for boundary value differential and integral problems: Theory and applications

In this work, we analyse the error of a discrete Petrov-Galerkin scheme for nonlinear mth-order ordinary differential and integrodifferential equations on a finite interval subject to nonlinear side conditions. As a trial space we chose high-order Cm-splines. We prove optimal-order convergence and superconvergence in the knots for lower-order derivatives, where the range of derivatives for these enhanced convergences to hold is determined by the behaviour of the nonlocal part of the integrodifferential equation. Our results extend and simplify earlier results by Ganesh and Sloan [1]. merical experiments in this work, for several singularly perturbed ordinary differential equations, demonstrate the power of our scheme that does not require any mesh restriction.