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


Authors: M. C. Delfour and F. Dubeau
Journal: Math. Comp. 47 (1986), 169-189, S1
MSC: Primary 65L20
DOI: https://doi.org/10.1090/S0025-5718-1986-0842129-0
MathSciNet review: 842129
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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}$-norm is $ K + 1$. The nodal-convergence rate can go up to $ 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.


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DOI: https://doi.org/10.1090/S0025-5718-1986-0842129-0
Article copyright: © Copyright 1986 American Mathematical Society