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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.

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
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