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This paper studies the approximation of the solution of nonlinear ordinary differen- tial equations by (discontinuous) piecewise polynomials of degree K and traces at the nodes of discretization. A mesh-dependent variational framework underlying this discontinuous ap- proximation is derived. Several families of one-step, hybrid and multistep schemes are obtained. It is shown that the convergence rate in the L2-norm is K + 1. The nodal-conver- gence rate can go up to 2 K + 2, depending on the particular scheme under consideration. The mesh-dependent variational framework introduced here is of special interest in the approxi- mation of the solution of optimal control problems governed by differential equations. 1. Introduction. The object of this paper is the study of (discontinuous) piecewise polynomial approximations to the solution of systems of nonlinear ordinary dif-

Discontinuous Polynomial Approximations in the Theory of One-Step, Hybrid and Multistep Methods for Nonlinear Ordinary Differential Equations