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

Delfour, M. C.; Dubeau, F.

doi:10.1090/S0025-5718-1986-0842129-0

Discontinuous polynomial approximations in the theory of one-step, hybrid and multistep methods for nonlinear ordinary differential equations
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by M. C. Delfour and F. Dubeau PDF

Math. Comp. 47 (1986), 169-189 Request permission

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.

References

I. Babuška and J. Osborn, Analysis of finite element methods for second order boundary value problems using mesh dependent norms, Numer. Math. 34 (1980), no. 1, 41–62. MR 560793, DOI 10.1007/BF01463997
I. Babuška, J. Osborn, and J. Pitkäranta, Analysis of mixed methods using mesh dependent norms, Math. Comp. 35 (1980), no. 152, 1039–1062. MR 583486, DOI 10.1090/S0025-5718-1980-0583486-7
F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. R-2, 129–151 (English, with French summary). MR 365287
J. C. Butcher, A modified multistep method for the numerical integration of ordinary differential equations, J. Assoc. Comput. Mach. 12 (1965), 124–135. MR 178573, DOI 10.1145/321250.321261
J. C. Butcher, Coefficients for the study of Runge-Kutta integration processes, J. Austral. Math. Soc. 3 (1963), 185–201. MR 0152129
J. C. Butcher, Implicit Runge-Kutta processes, Math. Comp. 18 (1964), 50–64. MR 159424, DOI 10.1090/S0025-5718-1964-0159424-9
J. C. Butcher, Integration processes based on Radau quadrature formulas, Math. Comp. 18 (1964), 233–244. MR 165693, DOI 10.1090/S0025-5718-1964-0165693-1
J. C. Butcher, An algebraic theory of integration methods, Math. Comp. 26 (1972), 79–106. MR 305608, DOI 10.1090/S0025-5718-1972-0305608-0

Vorlesungen über relle Funktionen

Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174

Sur l’Approximation des Équations Différentielles Linéaires par des Méthodes de Runge-Kutta

M. C. Delfour, The linear quadratic optimal control problem for hereditary differential systems: theory and numerical solution, Appl. Math. Optim. 3 (1976/77), no. 2-3, 101–162. MR 444189, DOI 10.1007/BF01441963

Piecewise Discontinuous Polynomial Approximation of Nonlinear Ordinary Differential Equations

M. Delfour, W. Hager, and F. Trochu, Discontinuous Galerkin methods for ordinary differential equations, Math. Comp. 36 (1981), no. 154, 455–473. MR 606506, DOI 10.1090/S0025-5718-1981-0606506-0
M. C. Delfour and F. Trochu, Discontinuous finite element methods for the approximation of optimal control problems governed by hereditary differential systems, Distributed parameter systems: modelling and identification (Proc. IFIP Working Conf., Rome, 1976) Lecture Notes in Control and Information Sci., Vol. 1, Springer, Berlin, 1978, pp. 256–271. MR 0502085
Michel Defour and François Trochu, Approximation des équations différentielles et problèmes de commande optimale, Ann. Sci. Math. Québec 1 (1977), no. 2, 211–225 (French). MR 482496

Approximations Polynômiales par Morceaux des Équations Différentielles

Finite Element Solution of Transposed Boundary Value Problems

C. W. Gear, Hybrid methods for initial value problems in ordinary differential equations, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (1965), 69–86. MR 179490
William B. Gragg and Hans J. Stetter, Generalized multistep predictor-corrector methods, J. Assoc. Comput. Mach. 11 (1964), 188–209. MR 161476, DOI 10.1145/321217.321223
Preston C. Hammer and Jack W. Hollingsworth, Trapezoidal methods of approximating solutions of differential equations, Math. Tables Aids Comput. 9 (1955), 92–96. MR 72547, DOI 10.1090/S0025-5718-1955-0072547-2
Bernie L. Hulme, Discrete Galerkin and related one-step methods for ordinary differential equations, Math. Comp. 26 (1972), 881–891. MR 315899, DOI 10.1090/S0025-5718-1972-0315899-8
Bernie L. Hulme, One-step piecewise polynomial Galerkin methods for initial value problems, Math. Comp. 26 (1972), 415–426. MR 321301, DOI 10.1090/S0025-5718-1972-0321301-2
Vladimir Ivanovich Krylov, Approximate calculation of integrals, The Macmillan Company, New York-London, 1962, 1962. Translated by Arthur H. Stroud. MR 0144464
P. Lasaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974) Publication No. 33, Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 89–123. MR 0658142
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243

Schéma d’Approximation pour des Équations Integro-Différentielles de Riccati

J. T. Oden and L. C. Wellford Jr., Discontinuous finite element approximations for the analysis of acceleration waves in elastic solids, The mathematics of finite elements and applications, II (Proc. Second Brunel Univ. Conf. Inst. Math. Appl., Uxbridge, 1975) Academic Press, London, 1976, pp. 269–285. MR 0483946
P.-A. Raviart and J. M. Thomas, Primal hybrid finite element methods for $2$nd order elliptic equations, Math. Comp. 31 (1977), no. 138, 391–413. MR 431752, DOI 10.1090/S0025-5718-1977-0431752-8
Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR 0443377

Sur l’Analyse Numérique des Méthodes d’Éléments Finis Hybrides et Mixtes

RAIRO Sèr. Rouge

L. C. Wellford Jr. and J. T. Oden, Discontinuous finite-element approximations for the analysis of shock waves in nonlinearly elastic materials, J. Comput. Phys. 19 (1975), no. 2, 179–210. MR 403389, DOI 10.1016/0021-9991(75)90087-x
L. C. Wellford Jr. and J. T. Oden, A theory of discontinuous finite element Galerkin approximations of shock waves in nonlinear elastic solids. I. Variational theory, Comput. Methods Appl. Mech. Engrg. 8 (1976), no. 1, 1–16. MR 489272, DOI 10.1016/0045-7825(76)90049-9
L. C. Wellford Jr. and J. T. Oden, A theory of discontinuous finite element Galerkin approximations of shock waves in nonlinear elastic solids. I. Variational theory, Comput. Methods Appl. Mech. Engrg. 8 (1976), no. 1, 1–16. MR 489272, DOI 10.1016/0045-7825(76)90049-9