Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems

Pages: 2393 - 2416, Volume 11, Issue 6, November 2012

doi:10.3934/cpaa.2012.11.2393       Abstract        References        Full Text (529.8K)       Related Articles

Maurizio Grasselli - Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi, 9, I-20133 Milano, Italy (email)
Morgan Pierre - Laboratoire de Mathématiques et Applications UMR CNRS 6086, Université de Poitiers, Téléport 2 - BP 30179, Boulevard Marie et Pierre Curie, 86962 Futuroscope Chasseneuil, France (email)

Abstract: Following a result of Chill and Jendoubi in the continuous case, we study the asymptotic behavior of sequences $(U^n)_n$ in $R^d$ which satisfy the following backward Euler scheme:

$\varepsilon\frac{(U^{n+1}-2U^n+U^{n-1}}{\Delta t^2} +\frac{U^{n+1}-U^n}{\Delta t}+\nabla F(U^{n+1})=G^{n+1}, n\ge 0, $

where $\Delta t>0$ is the time step, $\varepsilon\ge 0$, $(G^{n+1})_n$ is a sequence in $ R^d$ which converges to $0$ in a suitable way, and $F\in C^{1,1}_{l o c}(R^d, R)$ is a function which satisfies a Łojasiewicz inequality. We prove that the above scheme is Lyapunov stable and that any bounded sequence $(U^n)_n$ which complies with it converges to a critical point of $F$ as $n$ tends to $\infty$. We also obtain convergence rates. We assume that $F$ is semiconvex for some constant $c_F\ge 0$ and that $1/\Delta t

Keywords:  Łojasiewicz inequality, gradient-like systems, backward Euler scheme, proximal method
Mathematics Subject Classification:  Primary: 65L20, 65L12; Secondary: 65M12, 65M60

Received: October 2010;      Revised: December 2010;      Published: April 2012.

 References