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• W2335201328 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: $varepsilonfrac{(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}, nge 0, $ where $Delta t>0$ is the time step, $varepsilonge 0$, $(G^{n+1})_n$ is a sequence in $ R^d$ which converges to $0$ in a suitable way, and $Fin C^{1,1}_{l o c}(R^d, R)$ 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 $c_Fge 0$ and that $1/Delta t<c_F/2$; in the case $varepsilon=0$, these last two assumptions can be dropped off if the scheme is defined by a minimization algorithm. Applications to space and time discretizations of the damped wave equation and of the modified Swift-Hohenberg equation are given." @default.
• W2335201328 created "2016-06-24" @default.
• W2335201328 creator A5054462385 @default.
• W2335201328 creator A5082742339 @default.
• W2335201328 date "2012-04-01" @default.
• W2335201328 modified "2023-10-06" @default.
• W2335201328 title "Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems" @default.
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• W2335201328 doi "https://doi.org/10.3934/cpaa.2012.11.2393" @default.
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