Discontinuous Galerkin methods for ordinary differential equations

Delfour, M.; Hager, W.; Trochu, F.

doi:10.1090/S0025-5718-1981-0606506-0

Discontinuous Galerkin methods for ordinary differential equations
HTML articles powered by AMS MathViewer

by M. Delfour, W. Hager and F. Trochu PDF

Math. Comp. 36 (1981), 455-473 Request permission

Abstract:

A class of Galerkin methods derived from discontinuous piecewise polynomial spaces is analyzed. For polynomials of degree k, these methods lead to a family of one-step schemes generating approximations up to order $2k + 2$ for the solution of an ordinary differential equation.

References

Ivo Babuška, Error-bounds for finite element method, Numer. Math. 16 (1970/71), 322–333. MR 288971, DOI 10.1007/BF02165003

The Mathematical Foundation of the Finite Element Method with Application to Partial Differential Equations

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, 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
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 Opérationnelles 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. 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

Discontinuous Approximation of Ordinary Differential Equations and Application to Optimal Control Problems

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

Mathematical Aspects of Finite Elements in Partial Differential Equations

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
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
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