Discontinuous polynomial approximations in the theory of one-step, hybrid and multistep methods for nonlinear ordinary differential equations

Abstract

This paper studies the approximation of the solution of nonlinear ordinary differential equations by (discontinuous) piecewise polynomials of degree K and traces at the nodes of discretization. A mesh-dependent variational framework underlying this discontinuous approximation is derived. Several families of one-step, hybrid and multistep schemes are obtained. It is shown that the convergence rate in the L 2 {L^2} -norm is K + 1 K + 1 . The nodal-convergence rate can go up to 2 K + 2 2K + 2 , depending on the particular scheme under consideration. The mesh-dependent variational framework introduced here is of special interest in the approximation of the solution of optimal control problems governed by differential equations.