Discontinuous Galerkin methods for ordinary differential equations

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.

[1]  B. Hulme.  Discrete Galerkin and related one-step methods for ordinary differential equations , 1972 .

[2]  J. Butcher.  Implicit Runge-Kutta processes , 1964 .

[3]  I. Babuška,et al.  Analysis of finite element methods for second order boundary value problems using mesh dependent norms , 1980 .

[4]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[5]  G. Dahlquist.  A special stability problem for linear multistep methods , 1963 .

[6]  John C. Butcher,et al.  An algebraic theory of integration methods , 1972 .

[7]  J. T. Oden,et al.  Discontinuous finite-element approximations for the analysis of shock waves in nonlinearly elastic materials , 1975 .

[8]  B. Hulme.  One-step piecewise polynomial Galerkin methods for initial value problems , 1972 .

[9]  John C. Butcher,et al.  Integration processes based on Radau quadrature formulas , 1964 .

[10]  Michel C. Delfour,et al.  The linear quadratic optimal control problem for hereditary differential systems: Theory and numerical solution , 1976 .

[11]  P. Raviart,et al.  On a Finite Element Method for Solving the Neutron Transport Equation , 1974 .

[12]  I. Babuska,et al.  Analysis of mixed methods using mesh dependent norms , 1980 .

[13]  OnA-stable implicit Runge-Kutta methods , 1977 .

[14]  B. L. Ehle.  A-Stable Methods and Padé Approximations to the Exponential , 1973 .

[15]  A. Aziz.  The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations , 1972 .

[16]  O. Axelsson.  On the efficiency of a class of a-stable methods , 1974 .

[17]  M. C. Delfour,et al.  Discontinuous finite element methods for the approximation of optimal control problems governed by hereditary differential systems , 1978 .

[18]  G. Strang,et al.  An Analysis of the Finite Element Method , 1974 .

[19]  J. T. Oden,et al.  A theory of discontinuous finite element galerkin approximations of shock waves in nonlinear elastic solids part 1: Variational theory , 1976 .

[20]  H. A. Watts,et al.  A-stable block implicit one-step methods , 1972 .

[21]  I. Babuska.  Error-bounds for finite element method , 1971 .

[22]  J. T. Oden,et al.  A theory of discontinuous finite element galerkin approximations of shock waves in nonlinear elastic solids part 2: Accuracy and convergence , 1976 .

[23]  J. M. Watt.  Numerical Initial Value Problems in Ordinary Differential Equations , 1972 .

[24]  J. Butcher.  Coefficients for the study of Runge-Kutta integration processes , 1963, Journal of the Australian Mathematical Society.