##
**The dynamical behavior of the discontinuous Galerkin method and related difference schemes.**
*(English)*
Zbl 0998.65080

The authors study the behavior of the discontinuous Galerkin finite element method for initial value problems in ordinary differential equations which satisfy two types of dissipativity assumptions. Denoting by (1) \( y'(t)= f(y(t),t)\), \(y(t) \in {\mathbb{R}}^d,\) the differential equation, the authors consider first the so called contractivity assumption: \( ( f(u,t)-f(v,t), u-v) \leq - c \|u - v \|^2 \) for all \( u, v \in {\mathbb{R}}^d \) and \( t \geq 0 \) where \(c\) is a positive constant and \( (\;,\;)\) is an inner product in \( {\mathbb{R}}^d \). Note that this assumption implies that for any two solutions of (1), \( \exp ( c t) \|u(t) - v(t) \|\) is a non increasing function of \(t\) for all \( t \geq 0 \). The second dissipativity condition requires that \( ( f(v,t), v) \leq \alpha (t) - \beta (t) \|v \|^2 \) for all \( v \in {\mathbb{R}}^d\), \( t \geq 0 \), where \( \alpha \) and \( \beta\) are non negative and positive functions with \( \sup \{ \alpha(t)/\beta(t)\); \(t \geq 0 \} = R^2>0\) and \( \lim_{T \to \infty} \int_0^T \beta (s) ds = \infty\). Here the vector field of (1) points inwards on balls of sufficiently large radius and if \(f\) is locally Lipschitz continuous, for all \( \varepsilon > 0 \) and \( y_0\) the solution of (1) such that \( y(0)=y_0\) remains in the ball of radius \( R + \varepsilon \) for \(t\) sufficiently large.

After introducing in section 3 the discontinuous Galerkin (dG) finite element method for (1) and discuss the existence and uniqueness of the corresponding approximation, the authors study in section 4 the approximation properties of the discrete semigroup defined by the dG method proving in Th. 4.1 that the main properties of the continuous time semigroup translate to the discrete time counterpart. Next in section 5 two main theorems which prove the preservation of contractivity and dissipativity in the above sense are given together with additional comments on the continuous and discrete constants.

Finally in section 6 the effect of the use of an approximate quadrature on the dissipativity behavior is considered. Here two natural conditions on the quadrature are introduced that guarantee the dissipativity. Some examples are presented to show that for particular quadratures the resulting dG methods are equivalent to non confluent BN stable Runge-Kutta schemes.

After introducing in section 3 the discontinuous Galerkin (dG) finite element method for (1) and discuss the existence and uniqueness of the corresponding approximation, the authors study in section 4 the approximation properties of the discrete semigroup defined by the dG method proving in Th. 4.1 that the main properties of the continuous time semigroup translate to the discrete time counterpart. Next in section 5 two main theorems which prove the preservation of contractivity and dissipativity in the above sense are given together with additional comments on the continuous and discrete constants.

Finally in section 6 the effect of the use of an approximate quadrature on the dissipativity behavior is considered. Here two natural conditions on the quadrature are introduced that guarantee the dissipativity. Some examples are presented to show that for particular quadratures the resulting dG methods are equivalent to non confluent BN stable Runge-Kutta schemes.

Reviewer: Manuel Calvo (Zaragoza)

### MSC:

65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

65L12 | Finite difference and finite volume methods for ordinary differential equations |

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

### Keywords:

discontinuous Galerkin finite element method; initial value problems; stability; numerical examples; dissipativity; quadrature; Runge-Kutta schemes
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\textit{D. J. Estep} and \textit{A. M. Stuart}, Math. Comput. 71, No. 239, 1075--1103 (2002; Zbl 0998.65080)

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### References:

[1] | Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. · Zbl 0383.65058 |

[2] | B. Cockburn, G. E. Karniadakis, and C.-W. Shu , Discontinuous Galerkin Methods: Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, vol. 11, Springer-Verlag, New York, 2000. |

[3] | K. Dekker and J. G. Verwer, Stability of Runge-Kutta methods for stiff nonlinear differential equations, CWI Monographs, vol. 2, North-Holland Publishing Co., Amsterdam, 1984. · Zbl 0571.65057 |

[4] | M. C. Delfour and F. Dubeau, Discontinuous polynomial approximations in the theory of one-step, hybrid and multistep methods for nonlinear ordinary differential equations, Math. Comp. 47 (1986), no. 175, 169 – 189, S1 – S8. With a supplement. · Zbl 0633.65068 |

[5] | M. Delfour, W. Hager, and F. Trochu, Discontinuous Galerkin methods for ordinary differential equations, Math. Comp. 36 (1981), no. 154, 455 – 473. · Zbl 0469.65053 |

[6] | K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Computational differential equations, Cambridge University Press, Cambridge, 1996. · Zbl 0946.65049 |

[7] | Donald Estep, A posteriori error bounds and global error control for approximation of ordinary differential equations, SIAM J. Numer. Anal. 32 (1995), no. 1, 1 – 48. · Zbl 0820.65052 · doi:10.1137/0732001 |

[8] | Donald J. Estep, Mats G. Larson, and Roy D. Williams, Estimating the error of numerical solutions of systems of reaction-diffusion equations, Mem. Amer. Math. Soc. 146 (2000), no. 696, viii+109. · Zbl 0998.65096 · doi:10.1090/memo/0696 |

[9] | D. Estep and S. Larsson, The discontinuous Galerkin method for semilinear parabolic problems, RAIRO Modél. Math. Anal. Numér. 27 (1993), no. 1, 35 – 54 (English, with English and French summaries). · Zbl 0768.65065 |

[10] | Donald A. French and Søren Jensen, Global dynamics of a discontinuous Galerkin approximation to a class of reaction-diffusion equations, Appl. Numer. Math. 18 (1995), no. 4, 473 – 487. · Zbl 0839.65102 · doi:10.1016/0168-9274(95)00093-A |

[11] | Adrian T. Hill, Dissipativity of Runge-Kutta methods in Hilbert spaces, BIT 37 (1997), no. 1, 37 – 42. · Zbl 0873.65058 · doi:10.1007/BF02510171 |

[12] | A. T. Hill, Global dissipativity for \?-stable methods, SIAM J. Numer. Anal. 34 (1997), no. 1, 119 – 142. · Zbl 0870.65073 · doi:10.1137/S0036142994270971 |

[13] | A. R. Humphries and A. M. Stuart, Runge-Kutta methods for dissipative and gradient dynamical systems, SIAM J. Numer. Anal. 31 (1994), no. 5, 1452 – 1485. · Zbl 0807.34057 · doi:10.1137/0731075 |

[14] | Claes Johnson, Error estimates and adaptive time-step control for a class of one-step methods for stiff ordinary differential equations, SIAM J. Numer. Anal. 25 (1988), no. 4, 908 – 926. · Zbl 0661.65076 · doi:10.1137/0725051 |

[15] | J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York-London, 1970. · Zbl 0241.65046 |

[16] | A. M. Stuart and A. R. Humphries, Dynamical systems and numerical analysis, Cambridge Monographs on Applied and Computational Mathematics, vol. 2, Cambridge University Press, Cambridge, 1996. · Zbl 0869.65043 |

[17] | Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988. · Zbl 0662.35001 |

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